Furries are known to be irrational, quick to anger, easily offended, self-righteous and very, very, very dim. Here are some tips and facts for dealing with and arguing with Furries that you may find useful. By knowing how someone will react, you have already won half the battle.
Speak calmly and clearly because...
Acting angrily towards a Furry will cause it to go into defensive mode and start ignoring everything you say, it's much better to stay calm and...
Carefully inform them that they are perverts. This is very difficult, convincing a Furry how absurd their subculture is is like masturbating with chopsticks.
Remember, some Furries are very wealthy. It may be financially prudent NOT to immediatley attack a Furry. Some Furries have been known to spend thousands and thousands of dollars on Furry paraphernalia, some Furries will even pay you to have internet sex with them. If all you have to do is type YIFF YIFF YIFF into an AIM window for 60 seconds in return for your Paypal account overflowing, then go for it.
Most Furries are obsessed with Japanese "anime" cartoons - while this is by no means exclusive to Furries, it is often a Furry interest due to many of these cartoons featuring Furry characters - it's very possible that the only human voices they hear for months on end come from the 3 people that do voiceovers for these childrens programmes. Pointing out the poor quality of these saturday-morning cartoons is a sure fire way to get a reaction, from "Akira" to "Ghost into the Shell" right up to "Dragonball Z."
Anyone that uses the term 'Furson' to describe themselves is too far gone to bother with.
Furryism is a disability.
George Bush and Jesus are the two things furries hate the most.
Furries have the sense of humor of a rock.
When cornered it is not unusual for Furries to start arguing in wolf calls. This is not a joke. If a Furry responds to you with something resembling "AROOOOO WOOOOOO AROOOO", you have successfully broken their spirit, good job.
Furries will often try and 'yiff' you when they're out of names to call you. This is really funny, because they honestly think it's a good way to make their opponent feel uncomfortable and thus cause them to quit arguing. At this point, you can safely claim to have broken their spirit. Well done.
Mudane is a Furry's favourite word, a Furry will almost always describe non-Furrys as Mundanes - it means 'ordinary' people. This has always puzzled me, it implies that non-Furrys are ordinary, average people, so this would make Furrys somehow special and unique in the world?

Maybe dressing up as a sexy Horse/Fish/Elephant nurse makes you somewhat unique when compared to the rest of humanity, but I would argue that it doesn't make you unique in any particularly positive or worthwhile way. Sticking your ♥♥♥♥ in a blender makes you unique, but people will still think you're a ♥♥♥♥♥♥♥ idiot for doing it. That is, unless you construct a subculture and lifestyle around sticking your ♥♥♥♥ in a blender, so you can associate purely with other people that stick their ♥♥♥♥♥ in blenders, thus deluding yourself into thinking that sticking your ♥♥♥♥ in a blender is the norm and not at all a crazy thing to do.
Furries will instantly assume that everything you know about Furries comes from various TV Documentaries about them that have been made over the past couple of years. Furries hate these documentaries and will make a point of mentioning them constantly, the fact of the matter is that there are no factual innacuracies in any of them, but Furries will tell you otherwise. You can use this as an advantage as they will automatically have a pre-packaged response to any argument based on the TV shows Furries have appeared in and will not be expecting you to use anything else.

The Furry mind works thuswise: 'Surely, 100% of the people who hate Furries must have been turned that way by THE EVIL ANTI-FURRY LIBERAL CONSERVATIVE GAY NAZI COMMUNIST JEW MEDIA, so if I carefully prepare a response to this and only this line of attack, I will always win... right?? RIGHT??'

No.
Furries will compare you to Hitler, since Furries have a persecution complex like no other group on the internet. There is a Usenet adage saying that whenever someone brings Hitler into a discussion, it's a sign that the discussion is over. Hitler is often Furries' first line of defence. There are several flaws in this tactic. Mocking someone that draws pictures of anthropomorphic animals pissing on each other is not morally equivalent to the crimes of the Nazi regime.

Furries are not a race or an ethnicity. Furries make a choice to join the Furry community and condone it's massive output of disgusting gunk.
"This is amusing" or "I find this amusing." These are both heavily overused furry memes, and furries trot out one or the other in an attempt to justify a hypersensitive reaction, or to try and get out of a tough spot where they look like idiots. For example, in an argument a furry might try to make light of the situation by saying "Oh, I don't really care, I find this amusing" in an attempt to psychologically place themselves above their opponents. Usually this comes just after they've been hilariously embarrassed in some way, or as a qualifier before or during some insane rant. Honestly, what do they expect to accomplish by this? a reaction of "Oh no! They find it amusing. Well, lets pack up and go home guys. We knew this day would come."?

Furries will inform you that they have the right to say, do and draw whatever they want and there's nothing you can do about it.

Remember, Furries DO have the right to say, do, write and draw whatever they want, but everyone else has those rights too, not just Furries.

I can mock Furries all I want, I can laugh at their stupid artwork, I can cackle at their poorly written poetry and stories, I can stare in awe at how much money they spend on Furcadia addons, I can post "HORSECOCK" in their Livejournals if I want to and I will speak to Furries in the harshest language I can muster and there's nothing they can really do. Furries often wave their freedom of expression around as an anything-goes pass to let them do whatever they want while avoiding criticism, but constantly forget that it also applies to others. They often seek to silence anyone that opposes them or mocks them, completely missing the irony of doing so. That brings us to the next point...
Furries will take special care to tell you how they have 'won' an argument at every turn, they will also twist words and use twisted logic to make themselves seem superior. Remember, Furries can never truely win an argument, since at the end of the day when everyone goes home, they are still Furries and you are not. In this sense, a Furry can never put forward a winning argument. A simple rebuff of any argument with "But you are a furry" is invincible.
When in trouble, Furries will often start to post modified Garfield cartoons in an attempt to mock you. Remember, arguing with a Furry is much like arguing with a small child, if you tell a child not to touch the fire, it won't completely understand what you're trying to say. You have to explain why the child shouldn't touch the fire in language that the child's simple mind can understand. Either that, or scream at the child until it is reduced to a sobbing wreck. Either way works well.
Another often-used tactic is to deflect the blame of all of the terrible and awful things about the Furry community onto a small, anonymous group referred to as 'Furverts' (Basically, Furries into all of that nasty stuff). While it's true that not ALL Furries take delight in reading stories about gay threesomes featuring Disney characters, a majority do so.