Where do you live? European Pyrex is made from borosilicate glass, the same as in laboratory's equipment; American Pyrex is made from common soda-lime glass.
If you are in America, don't bother trying it at all; soda-lime glass is sensitive to thermal shock. Even though it's tempered for kitchenware, it is nowhere near good enough for the burner.
In Europe, you could take your chances if you have a bowl you don't mind risking. However, there is still a significant chance that it will break on the burner some day. While I think that they use the same raw material for both kitchen dishes and laboratory test tubes (which are obviously OK on a gas burner), kitchen stuff is much thicker. This makes it much more likely to break under thermal expansion.
If you decide to make the experiment with a borosilicate Pyrex, take care to warm it gradually, starting with a small flame, and don't pour cold ingredients into it. Proceed at your own risk. And ask yourself if you really have no pots better suited for the task.
I see two sides of this question. First, should you keep a redundant tool in your kitchen, and second, is there a specific purpose to this pot.
I may be biased because my kitchen is small, but my point of view is to get rid of any redundant clutter. Get a few pieces of quality cookware and use them every day. There is no sense in having 3 pots which do the same thing, even if one is half a liter bigger than the other one. (The worst situation is when the small one is decent, and the big one is bad quality, but sometimes you need the volume, so you still keep the big one around). If you don't use a thing, remove it from your kitchen. (Of course, donation/ebay is better than trash from a sustainability point of view).
The second question is, even if you don't use it now, is there an occasion where you might want to use it? It so happens that there is.
Generally, it is a bad idea to use Pyrex on the stove (see this question), and the handle makes the pot impractical to use in the oven (you don't give a size, but I bet you can't place it in the center, if you're able to close the oven door at all). But there is something else you can do with it. This thing makes an awesome double boiler.
If you don't have a professional double boiler, the usual way to "build" one at home is to find a bowl and a pot so that the upper rim diameter of the bowl is slightly bigger than the pot's diameter. You put water in the pot, put the bowl on the pot so that its bottom is suspended in the boiling water, and fill the bowl with your ingredients. It works with any moderately heat-resistant bowls (including glass, but don't use nylon), but I prefer to use glass.
First, glass is a bad heat conductor, which means that it heats more gradually and you have more time to work before your ingredients overheat. This makes it much better than steel. Second, glass is transparent. This means that you can see the boiling water and adjust the temperature according to how strong your boil is. You can't do this with a metal or ceramic bowl.
The drawback to this setup is that at some point, your ingredients are close to overheating. You have to remove the bowl from the heat, and do so quickly. Well, by then the bowl is very hot, it is slippery on the outside because of condensed steam, and your mittened hands slip on the rim withot findind purchase, while the hot steam penetrates through the cloth to cook your skin. The best solution would be a glass bowl with a handle, and that's just what you have here.
The problem in using the dish as a double boiler top is that its bottom is rather flat, making it less suited for small amounts of ingredients. Depending on how much you cook at once and how big the pot is, this may or may not an issue. The second problem is that you can't shock cool it to stop an uncontrolled overheating. But being as easy to grab as it is, you can just keep a cool bowl on the counter and dump the ingredients into it in case of emergency.
Conclusion: if you think you need a double boiler often enough to warrant the room occupied by this thing, you can keep it. But if you don't use it often, there is no point keeping it around as just another pot.
Best Answer
I have used borosilicate glass vessels on a number of different heat sources, both in laboratory circumstances and on a standard home gas stove.
Generally speaking, no. I mean, it depends on what you mean by "safe" -- borosilicate glass is less likely to shatter in explosive ways compared to some other types of glass. Glass vessels designed specifically for cooking are probably even less likely to fail in a way that is dangerous.
But will the vessel survive doing such things repeatedly over an extended period? Probably not. It may eventually crack. Depending on how extreme the temperature difference exposed to, it may even break more violently. But most likely, cooking vessels like this will simply fail by developing a crack (which will then leak contents).
I would consult the specific documentation that comes with any glass cooking vessel for guidelines appropriate to it. (And in case this isn't clear, I would NOT recommend using borosilicate glass vessels on the stove top unless they are specifically designed for such use. Even lab glassware will eventually crack if not used over an even heat source, which is why labs tend to use various heat diffusion devices when employing an open flame on a glass vessel.)
For example, the Pyrex company had a set of glass pots for stove top use called Flameware, which was discontinued (I believe) in the late 1970s. Instructions can be found here, which state (among other things):
All of these warnings have to do with avoiding stress created through rapid temperature changes, which will eventually cause a pot to fail. Note these are warnings for a line of Pyrex specifically designed for stove top cooking.
For a standard producer of borosilicate glass for stove top use today, see JENA's Trendglas, whose products here are specifically designed for direct heat:
These glass products have instructions here. And that document includes similar warnings to the Pyrex products discussed above:
Basically, glass is a substance that can crack from thermal shock. No amount of engineering is going to make a glass vessel as durable and resistant to thermal shock as a metal pot. That said, scientists use borosilicate glass all the time in complex procedures involving heating and cooling, and exploding glassware is rare unless you do something very stupid. But cracks and failures of glassware that is used over and over for applications involving thermal stress and rapid temperature changes are also to be expected.
If you follow the manufacturer's instructions for your pot and take advice like that listed above (avoiding thermal stress, heating gradually, etc.), some glass pots can last for many years or decades. And you may get away with occasionally forgetting and making a mistake. But even if cracks don't appear immediately, those stresses can build up over time and eventually cause failure.
EDIT: After reading over some of the comments that have appeared, I wanted to incorporate some ideas and re-emphasize a point I stated above: I would NOT recommend using borosilicate glass vessels on the stove top unless they are specifically designed for such use.
Some of the answers on previous questions about borosilicate versions of Pyrex (still to be found, particularly outside the U.S.) say it may be possible to attempt to heat borosilicate containers (Pyrex or otherwise) on the stove top, but I would strongly advise against this unless it's actually a vessel labeled for stove top use or described in its instructions as appropriate for stove top use.
It's not just borosilicate glass itself that makes a pot or other vessel safe for the stove top. Glass intended for use on the stove top usually has higher standards for production. It is often more even and thinner (like lab glassware) than some other types of glass bakeware and kitchenware, as thicker glass will have a larger thermal gradient between the surface and interior that can make the glass vulnerable to fracture. It may employ a somewhat different formulation of ingredients in the glass (to respond better to thermal stress, rather than to emphasize durability). It may also have specific design features in terms of its shape and variance in thickness to account for the type of expansion that occurs during stovetop heating (i.e., heating from the bottom) vs. bakeware that will usually expand more at the top (because food inside will be cooler in the initial stages of baking, while the top of a glass baking dish may get hotter without food in direct contact with it). And I'm sure there are other design features that make glass safer for stove top use.
User SiHa linked a video showing what can happen if borosilicate glassware is inappropriately heated. That measuring cup is specifically labeled as not for stove top use, and it's not designed for that sort of stress. A measuring cup is probably only expected to take the variation in temperature caused by things like introducing boiling water, not an open flame applied to it. But also note that the catastrophic failure in that video (and mild explosion) likely happens due to two things:
I don't mean to scare anyone too much here, only to emphasize the importance of only using glassware designed for direct heat on the stove top. As I mentioned in my original answer above, when borosilicate glassware designed for direct heating fails, it is more likely to crack (perhaps crack a lot, but still simply crack into pieces) if handled appropriately. That can create a mess, but it's usually not dangerous or explosive unless one does something stupid with it.