Use sunflower oil to make pesto

It's certainly safe. Butter, Oil, whatever, it's more about taste than anything else. Some people even push lard and butter as the "safe" options, because they require less processing than oil, and much less than margarine.

One thing which you should be aware of in regards to butter vs oil: butter has a significant amount of water in it, so you shouldn't substitute 1-to-1, or you will have a stronger flavour (and more fat) than perhaps you want.

If you're accustomed to salted butter, adding some salt will help ease the transition. If the flavour of olive oil is off-putting, there are plenty of more neutral flavoured oils that have a similar fat composition, or you can simply avoid extra virgin olive oil*.

*extra virgin olive oil in the U.S is largely a scam. In most of the world (countries that adhere to the IOC standard) this signifies "First Cold Press" olive oil. In the US it signifies "lamp grade olive oil with green food coloring". There are no restrictions on the use of terms like "extra virgin" in the US, so you can't guarantee that's what you're really getting just because it's on the label.

Coconut oil is a saturated fat, which makes it solid at room temperature. Above room temperature, it behaves exactly like any other oil and has a very high smoke point. You don't need to treat it in any special way.

I personally just use a knife or spoon to dig a little bit out of the container.

Coconut oil is considered to have a neutral flavour (with the exception of some "virgin" coconut oils). It's much weaker than, say, olive oil. You'll most likely not be able to detect any coconut flavour from oil alone - you have to use something like creamed coconut (a common ingredient in satays) for that.

P.S. Although it might appear to be similar to butter or margarine, keep in mind that it is actually a cooking oil, unlike butter which has milk solids that burn, or margarine which is hydrogenated to make it solid. Coconut oil is just barely solid and won't always work as a substitute for those other things.