Warning: Long post is long.
From Wikipedia, abridged and lightly opinionated for flavor; Felisha Day has been on several TV shows, several web serieseses including Dr. Horrible's Sing-Along Blog, a whole slew of movies I've never heard of, is on the Board of Directors for the International Academy of Web Television, (which sounds impressive) and last but not least voiced a character in at least one popular video game that I liked.
So yes, she is in fact relevant to quite a number of things and interests, (Board of Directors, dude.) it's just that most of them aren't video gaming. It is however unfair to say that she's a "glorified booth babe" (Board of Directors, dude.) just because she hasn't written for or designed a game of her own. Most "famous" video game people haven't. We can't all be Yahtzee, folks. The human population wouldn't last past the current generation.
Felecia Day is an actress, and a geeky one at that, she just also happens through no fault of her own to be a gorgeous redheaded ladyperson. This is not an isolated problem; There is this pervading logic that exists that attractive does not equal intelligent. This is untrue, and here is some algebra to help illustrate the point I'm trying to make;
(Hot =/= Smart) =/= Truth
I've really got to find a "does not equal" button for my keyboard... Anyway! Felicia Day being attractive doesn't detract from her qualification for geekdom, and we've got to get it out of our collective heads that it does. Need a reason? Here's a real simple one; WE'RE SCARING ALL THE HOT GEEKY GIRLS AWAY. You know that girl that you see everyday in where-ever that you want to spend all day talking about Bill Murray movies and that one Skyrim character you made that's based on Bill Murray? She might actually be interested and not want you to know because she'd be in for a world of shit if she admitted geekdom.
I'd like to think, however, that this guy who said these things, if he were not merely posting them on his personal Twitter thing, (I wouldn't say "private" because if he wanted privacy he wouldn't be on fucking Twitter,) if he were actually voicing these questions to Ms. Day, he'd have the presence of mind to word them differently. An example would be "How are you relevant to gaming? What, of merit, have you contributed to the medium? Have you set your sights on any noteworthy video game opportunities?" That's a far better way to ask the question, keeping the directness with none of the hostility, and including an opening for the subject to actually answer the fucking question. Although the big thing to remember here is that he was not actually posing these questions to Felicia Day. He wasn't actually saying them, he was just musing to himself. Personally, I regularly wish death and AIDS on everyone I share the road with, but I don't necessarily want to hurt anyone, I'm just venting frustration. I imagine that's what this guy was doing.
So, to sum up and conclude; No, a person should not be fired for their personal opinion, assuming of course that they can keep it to themselves.
Which this guy couldn't.