Can I leave ripe red tomatoes on the vine until I'm ready to use them?
How long do they stay good on the vine?
Will picking them cause them to go bad faster or does it make any difference?
It really depends on the variety. Some can stay on the vine for a long time after they're ripe (how long they can stay on is known as hang-time), and some have to be picked as soon as they're ripe. How picking affects the matter also seems dependent on the variety, as well as how ripe the fruit is when you pick it. It seems to me that if you pick them when they're still green they last longer after ripening. They don't have as much of a chance to split, crack, get eaten by wildlife and such, either, but this may affect the flavor.
When you pick tomatoes, some kinds are harder to pick than others (or take more force to pull off). For those, I recommend snipping them with scissors so you don't damage the plants or the fruit. Some tomatoes will come off very easily (either because the stem separates from the larger stem easily or because the calyx separates from the fruit easily. Even in this case, though, leaving the stem on can improve storage capacity and other things.
Some tomatoes are more prone to splitting and cracking than others, and this can affect the hang-time and the shelf-life.
You might be interested in RIN tomatoes (which stay firm/crisp even when fully ripe) and storage tomatoes, like Long Keeper. High anthocyanin tomatoes also have potential for increased shelf-life.
Some people prefer extremely ripe tomatoes (whether ripened indoors or on the vine), but I generally don't like then so ripe, since they get really soft and juicy; they lose their acidic flavor; they sometimes taste more like chlorine (to my tongue). I prefer to pick them either as soon as they ripen or within a few days, depending on the breed.
Here's a list of a few examples of how tomato breeds differed, for me: Black Beauty has great shelf-life and great hang-time. Fruits could be left on the vine for a really long time, and they'd stay good in storage for a really long time. Galapagos Island, however, can stay good on the vine for a decent amount of time (and progressively gets a deeper color long after ripening), but it's very prone to splitting when I pick it. Paul Robeson in 2014 for me would burst into liquid, if left on the vine more than a day after ripening, but if picked in good time had probably a regular shelf-life. Some varieties like Burpee Gloriana crack easily and can go bad easily for this reason. Some tomatoes look like they'd keep well, because they're firm and flawless, but they don't.
My source for terminology is just forums and other websites I've used for the last few years (people say hang-time in that context).
Stores and forums are another general source of information (not just for terminology).
I agree with Stormy about picking the tomatoes sooner than later if you want a bigger harvest. The same goes for okra and many peppers.
I have mixed feelings about what Stephie said about refrigeration. On the one hand, she's right (you don't have to do it, they do often keep a good while unrefrigerated, and the taste can be much improved). On the other hand, it really depends. Tomatoes can sometimes (not always) taste better (or at least different, in subjective ways) if refrigerated. I thought Valencia tasted better refrigerated, anyhow. For other varieties, there's a certain taste and texture that refrigeration can impart, and it may or may not be what you're looking for (in my case, I was looking for it, at times). Tomatoes will ripen a lot more slowly in the refrigerator, too, whether or not they stay good longer (so if you don't want them over-ripe, and you want to store them a while, keeping them cooler should help). However, they may be prone to contracting fruit rot pathogens in the refrigerator, due to produce that may have gone bad in the past in the refrigerator (so, a cold storage area like a cellar is probably much better for long-term storage).
Regarding Graham Chiu and Stormy's discussion:
Tomatoes are tender perennials often grown as annuals. Some kinds of tomatoes will grow and fruit continually (indeterminates are supposed to do this, although in my experience some are more prone to fruiting continually than others), and some will stop and die after the first wave (these are usually determinate, but not all determinate tomatoes do this). (Some kinds will do other things.)
I'm of the opinion that those that would die after the first wave will live and eventually fruit again if you prune them after harvest, though, provided they're healthy. As far as heat influencing ripening goes, I've heard often that it can hamper ripening, but I personally think something else that just often corresponds with the heat, is responsible instead, since I've seen tomatoes ripen just fine in very hot temperatures (and I've seen them struggle, too). The time of the season seems to impact ripening more than the temperature (same for green onion growth, if you harvest the greens and leave them to grow back), but I'm not sure what causes it. In my area, ripening and production seem to speed up considerably in mid August, which is usually when the temperature falls a few notches, but yeah, it can be hot then, too, and the garden still improves. I kind of feel like it's something about the sun.
Once they are red or ripe get them off the vine/bush! This promotes MORE fruit setting, ripening. Keep them cool and eat them sooner than later. Don't put in the refrigerator. Keep apples and bananas away from other vegetables or flowers you don't want them to ripen or die quickly.
Tomatoes are climacteric fruits so undergo a burst of respiration at the time of ripening, and then will continue to ripen after being removed from the vine. If the temperature is too hot ( over 85 deg F), this can inhibit the ripening process so you can remove the fruit and let it ripen inside instead which might give you a 5 day headstart.
However, your question is about fruit already ripe, and most people seem to think that vine ripened fruit have higher sugar levels. It shouldn't get more sugar from the vine once it's ripe so likely will make very little difference if you leave them on the vine.