The rules as written don’t allow for the possibility, the term “grease” covers a variety of substances, some of which don’t burn easily, and then regardless there is every reason to think that magically-created grease is not literally the same substance as any mundane grease and therefore would not share the properties not listed in the grease spell description.
On top of that, there is strong circumstantial evidence: in Mythic Adventures, there is an augmentation of the mythic version of grease that makes it flammable. 3.5 had a similar case, a 2nd-level incendiary slime spell that was “grease, but flammable.” These things wouldn’t exist if grease itself were already flammable.
Moreover, grease is one of the best spells for its level in the game, which makes many DMs leery of opening up yet more features for it. On the other hand, DMs also like to reward creativity, so some do allow grease to be lit on fire. You will have to ask your DM for a more solid answer than “the rules don’t say you can, so you can’t.” Or, if you're the DM, you should follow the line of reasoning you prefer to its conclusion.
My first character ever, a sorcerer, had grease as something of a signature spell, and this was one of the first “tricks” I tried with it. The DM didn’t allow it, and considering how much use I got out of the spell despite lacking that option, I think he made the right call, and I’ve made the same call in all of my games since. I have never seen anyone complain that grease is unusable without that feature. Again, it’s one of the best spells out there.
The answer is in your bold letters.
The subject can climb and travel on vertical surfaces or even traverse ceilings as well as a spider does. The affected creature must have its hands free to climb in this manner. The subject gains a climb speed of 20 feet; furthermore, it need not make Climb checks to traverse a vertical or horizontal surface (even upside down).
The description says that it doesn't need to make Climb checks in order to traverse a vertical or horizontal surface. In that case, the check is to overcome the slippery effect of Grease spell. I would assume that it is the same as saying that a creature with walking speed needn't make any Balance checks to walk on an horizontal surface. But it still needs to make them in order to overcome obstacles. Grease is an obstacle, not a surface.
The spell calls for a Balance check, but given the specificity of the situation you describe, your GM could perfectly call for a Climb check instead. It seems reasonable.
Best Answer
No, Grease (as per the spell), is not flammable.
There is already a spell that does this:
They would not have created a level two spell that specifically calls out being flammable as the distinction if the level one spell were already flammable.