White Collar fans have been known to come to virtual blows over Hilarie Burton's character Sara Ellis. Half of them think she's a strong, independent woman who is just too much for misogynist fangirls who hate seeing Neal Caffrey locking lips with another girl. The other half think she's an annoying cliché who only exists to give the main character a love interest. Don't want to take my word for it? Just check out the comments on this thread.
Kirk vs Picard is a classic that shows up less often now. If only because people prefer to complain about Archer and Janeway.
Then there are those who think that Sisko was the greatest captain EVAR and refuse to accept any and all Star Trek that isn't completely serialized like Deep Space Nine was.
A specific example from Doctor Who: the Flame War that resulted after it was announced that Christopher Eccleston was only doing one series as the Ninth Doctor got so out of control the entire Outpost Gallifrey forum had to be shut down for two days. Matters weren't helped when prominent fan Ian Levine, in what can only be described fairly as a moment of hysteria, elected to describe Eccleston as 'lower than a cockroach'.
In general, discussions about whether the New Series is better than the Classic Series or vice versa have a good probability of ending up as one of these.
Really, anything about Doctor Who will result in a flame war. Old series v. New Series, shipping battles, best Doctor, best companion, best episode, best season, best writer, best producer, worst Doctor, worst companion, worst episode, worst season, worst writer, worst producer, romance in the TARDIS, too much emotion, not enough emotion, stand alone episodes v. arcs, the TARDIS interior design, the TARDIS exterior design... and that's not even touching the Expanded Universe.
Also, word of advice (oh boy...): Do NOT mention the whole Tennantvs.Smith debate. You really don't want to encourage the crazies. There has been a lot of arguing on social networks about this.
God help you if you even DARE to mention the Joel vs Mike argument from Mystery Science Theater 3000 fandoms. The sheer amount of flames you will receive can be used to toast marshmallows.
Interestingly, study of one of the most active boards during the switch (and, more importantly, one of the few to have archived their posts), Usenet group alt.tv.mst3k, reveals comparatively little flame warring over Joel/Mike. It seems that, for at least one board, much more anger and vitriol was spilled over whether certain elements of the show were sexist and if Jim Mallon had been instrumental in "forcing" Joel out.
Visit a Supernatural fan community and mention that you like Ruby 2.0/Lisa/any chick that's been within 50 feet of Jensen Ackles and Jared Padalecki, up to and including their wives. Every hair you have will be singed off.
Glee fandom has this in spades—just trying getting into a "Kurtofsky" (Kurt/Karofsky) vs. "Klaine" (Kurt/Blaine) argument. If you prefer the former, you will be told that you clearly support bullying, hatred and sexual assault in Real Life. If the latter, you will be told you have no taste and only like boring Mary Sues and have just jumped on the Darren Criss bandwagon to seem cool. It's like the first twenty minutes of Saving Private Ryan, except a lot more gay.
Buffy the Vampire Slayer was well known for Buffy/Spike vs Buffy/Angel ship wars that got into flame war territory on occasion.
Battlestar Galactica - was the 1978 series better than the 2004 series? was Glen Larson a visionary or merely capitalizing on Star Wars? Was Ron Moore a hack just using another's idea or a gifted writer who breathed life into an old franchise? (And don't dare inquire about which Starbuck was better...) The extremes use the term "GINO" - Galactica In Name Only to refer to the series they don't like. Some forums are dominated by one group and will shout out anyone who happens to come by with the wrong preference. Fans do exist who like both series - they are either ignored or considered unenlightened morons by the more militant. About the only thing BSG fans CAN agree on is that Galactica1980 was terrible.