I answered yes. I put a UV filter on every lens that will take it, and my 70-200 f/2.8L IS II is no exception.
1) Most L lenses aren't fully weatherproof without it.
2) It's easier to clean (flat glass, no ribs/ridges, you're not brushing the front element)
3) The usual protection reasons... banging it against something, sand/dust, etc.
edited to add: LOL at the cling wrap response!!
1) Isn't true, almost all L lenses that are weather sealed do not need a filter to seal them, the 16-35 MkI and II and the 17-40 are the most notable exceptions.
3) Works great in theory, until you break the comparatively flimsy filter and rub nice shards of glass on your front element.
There are very good reasons for using filters, and equally valid reasons to not use them, it really is personal preference as lenses have been protected, and ruined, going either way.
I tend to use them in very harsh conditions (I am often in salty spray and sandy conditions) but the rest of the time leave them off as I always use hoods and doing so mitigates many of the reasons people give for using filters.
How about at a crazy reception party? Where a happy patron swings around a beer, slathering your 50L with its lens hood on full of bubbly joy? Well, in my case my filter was pretty ugly and require a moment of serious cleaning. I wouldn't have felt comfortable cleaning my front element as harshly as it needed to be cleaned.
Now, this wasn't a "harsh" environment at all but one of the many times which a filter has saved my lens.
No, personally I don't care about drinks, kids sticky fingers or any number of other things, modern lens coatings are pretty tough, but like I said, filter use is an entirely personal opinion, there are pluses and minuses to both sides. I have lost two 17" MacBook Pros to drinks, I have never degraded any lens element because of it. Say he had swung his glass enough to break your filter? In that situation he might have knocked my lens hood off, I can put it back on again whereas you have glass shards all over your front element and in your filter thread.
As for cleaning, I make my own solution out of Isopropyl Rubbing alcohol, ammonia solution and a little distilled water, it costs a couple of dollars for a good quantity and you can mix it strong to get ride of some very severe looking marks that often turn out to be grease and grime.
1. I always use lens hoods but that won't always save your lens or front element.
2. The filter is a second line of protection. As for shattering glass, I've had one shatter on my 24L II and simply unscrewed it, and continued to shoot the rest of the event. If I didn't have one, There would be one less 24L II in the world.
3. In the heat of good shots, stopping to carefully clean beer off your front element shows a lack of preparation as a candid-man and your in-experience to your employer.
Did you get those shots? You stopped because of that? Wheres your Filter