If I don't clean the rosin off my strings from time to time how will that affect the sound the strings make as the rosin builds up? How will it affect the playability, the way the bow moves over and catches the string?
There are a couple of things that can go wrong here if you don't clean off your rosin
Your bow needs to be able to grab the string. Does caked-on-rosin make it harder for your bow to pull the string, after a certain point? Perhaps. The interference can't be that great. After all, we often play, in orchestra rehearsals, for hours, without ever wiping the rosin off. Therefore, there must be a reasonable balance.
Not So Simple
The tone produced by your violin has a great deal to do with the "overtones of the harmonic sequence." There is a lot going on when you pull on your violin string. Not only is the whole string vibrating, but smaller portions of the string vibrate as well, and at different frequencies. Some of these portions are very, very small. If there are any irregularities in your string, it can prevent or disrupt these small vibrations. An irregularity could be a rust spot. I once bought a brand new set of strings that "whistled" when I used a certain contact point because rust stopped a certain set of vibrations from forming, leaving only the higher-pitched ones. This is also why we change our strings if we have been playing on them a long time, even if they look fine. Badly caked-on resin will have a similar dampening effect, diminishing the sound of your instrument.
That being said, it would take a bit to get to this point. One must be reasonable. I have always been of the mind to wipe off my strings when I am done practicing for the day. That seems to do it.
Hope this helps!
If you don't wipe the rosin off after each time you play, it will build up and eventually get a bit sticky so that the bow won't guide so smoothly. Dirty strings will also sound less open and fresh. From time to time, you should also loosen each string separately and rub it up and down with a clean cloth and something like meths. That should remove all the accumulated dirt. By the way, it's also good to loosen your bow from time to time and give the hairs a good brush with an old toothbrush.
Just to add to the other answers to say that I found a very useful thread on Violinist.com.
The most useful contribution (from my perspective) came from Michael Darnton who said this:
Too much rosin makes a lot of noise. Generally you want to use the minimum you can get away with. If you're talking about build-up, you may be using too much. Also, rosin on the strings makes noise, and you want to make sure to keep them clean. Steel wool is good for this. If you think you have too much on your bow (and I bet you do) stop rosining for a while, until you really need to, and wipe off all the extra you can right now.
If you think the hair is a problem, and you can't get to a rehair, hair can be cleaned with alcohol. Remove every last bit of rosin and start over. Be very careful not to get any on the wood, and leave it to dry completely (out of the case!) before you re-rosin. The hair under the rosin will be good: hair does not "wear out"--the reason for bow rehairs is packed and caked rosin, and the uneven hair that results from the hairs on one side of the bow breaking more than the other side.
Cleaning the rosin off the strings using steel wool is a bit too extreme for me, I'll stick to using a cloth, but I will try out his suggestion of cleaning the bow hair using alcohol and I will stop just rosining automatically every time I practice.