Is there any reason I should finance my car for 36 or 48 months instead of 60 months? Is anything wrong with financing for 60 months?

Let's take your questions one at a time:
>Is there any reason I should finance my car for 36 or 48 months instead of 60 months?<
Yes, there could be several. (1) You will generally pay less interest on a 36 or 48 month loan than you would on a 60 (assuming that we are not talking about 0% interest deals here).
For example, if you were financing $30,000 at 1.9% interest you would pay interest as follows:
36 months - $886.87
48 months - $1,178.23
60 months - $1,471.26
So, while your payments will be higher the shorter the term, your total interest paid will be lower.
(2) If you plan to get a new car every 3-4 years, you would probably want to have it as close to paid off as possible during that time. (3) You officially own the car sooner. (4) A longer period of time where you don't have to make car payments.
>Is anything wrong with financing for 60 months?<
As long as you plan on keeping the car for a while (say at least 7 or 8 years), and the interest rate isn't substantially higher, I would say not really. I would argue that it is reasonable to expect a useful life of at least 8-10 years on pretty much any new car these days (unless you treat it poorly and don't take care of it and/or you put 30,000+ miles on it every year, or something like that), so paying it off over 5 years is not necessarily that bad of a thing.
Just be aware that in most cases, you will pay more in interest for the car than on a shorter loan. You also may want to consider GAP insurance depending on how much you put down. If you don't put much down and finance it for 60 months, then there will be a pretty lengthy period of time (probably at least 2 and maybe even around 3 years) where you will probably owe more on the car than it is worth, so GAP insurance may be another cost you need to factor in.
Another point is that the interest rate on a 60 month loan may be higher than on a 36 or 48 month loan. That is not always the case, but it can be, so be sure to check on that before signing, because if the 60-month interest rate is higher, then the difference in interest paid would be even larger.
If you plan on getting a new car every 3 years or something like that, then I would probably suggest staying away fro ma 60-month loan.
While I typically think financing a car for 60-months is not always a bad thing, I would definitely NOT go any longer than that. Car dealers these days are all too happy to stretch out the terms to 72 and even 84 months to get the payment you want. All that does is put more money in the finance company's pocket and mean you're paying off your car for 6 or 7 years.
All in all, I think that you should strive to use a 36 or 48 month loan because you will pay less interest and it will "help you" buy a car that you can better afford. However, while I'm not necessarily opposed to 60-month auto loans in most cases, I would typically suggest that 60-months should be the longest auto loan that you consider.
Good luck.