Like so many others I broke the UV filter on my Canon lens, an EF 70- 300mm f/4-5.6 IS USM. I'm finding it difficult to remove the UV filter as the threads seem to have been warped with the breakage impact. Any suggestions?
If the rubber band or filter wrench doesn't work, the next step is a precision band saw.
If Adam Savage (the guy in the video) and his band saw is not available, you could also use a needle file or hacksaw to create a couple of notches in the front of the filter ring.
Once you've cut a couple of notches in the filter ring, you can use a steel rule to twist it off.
If the filter ring is too distorted to twist it off with the metal rule, you're going to need to remove as much of the filter glass as you can and then peel it off with a pair of (locking) needle-nose pliers. It may be easier if you cut all the way through the filter ring. Yes, that will nick the lens' filter threads. But if they are already warped they're going to need to be replaced anyhow.
When using a file, hacksaw, needle-nose pliers, etc. on the filter ring, it's not a bad idea to place a soft cloth over the front element of the lens to minimize the risk of scratching it.
If the lens hit something hard enough to warp the filter ring, the front element of the lens and filter threads are the least of your worries. The internal lens alignment needs to be checked.
For more about the overall subject of To filter or not to filter (for lens 'protection'), that is the question, please see this answer to is uv filter a must? here at Photography at Stack Exchange. There are lots of links to a plethora of questions/answers here and resources elsewhere on the subject. While there are times and places (sand/dust storms, at the beach, industrial settings with hot metal particles flying off grinders, etc.) where filters are useful for protecting the front of the lens, in catastrophic incidents such as yours they tend to be liabilities more often than not. A lens hood is normally better impact protection without any of the optical penalties of a flat glass filter.
On the other hand, when shooting in daylight, UV from the sunlight passing through your lens will inhibit the growth of fungal spores that may be (hint: are) in your lens¹. The filter stack directly in front of the sensor has a fairly strong UV filter anyway, so there's no real benefit to preventing UV light from entering your lens unless it is constantly exposed to sunlight 24/7/365, and there's a definite benefit to at least periodically allowing UV light through the lens.
¹ Fungal spores are everywhere on the surface of the earth, including the skin of most animals and the leaves of most plants. They require three things to multiply: Moisture, organic matter (provided by almost every kind of dust), and the absence of UV light.
If you can pick up a filter wrench that generally works well.
In a pinch, I find I can get enough traction by putting a rubber band around the filter then unscrewing the filter using the rubber band as a grip. This works especially well if you can slightly overhang the rubber band over the grip ridges on the filter ring.
If the damage was enough to warp the filter rings you might need to get the lens checked...