Do professors need to use up their grant money in a fixed period of time? Just curious, do professors need to use up their grant money in a fixed period of time?

Or can they reuse the left over amount for other projects?

What are the incentives for them to use grant money as opposed to "saving" them?

Thanks.
That depends on where the money comes from, but in general yes, they should use it. Just like everywhere else, budgets are tight. And if the money is not fully used in a year, the funder might assume that this professor can do well with less money, so the next year the funds are decreased. I witnessed a case a few years back where a professor decided it was time to get a fancy notebook for conferences in mid December, because there was still money left...

Another possibility is that some other institute/chair/professor puts his hands on the money if the professor does not keep a tight eye on it. I myself got asked once to "please move to this office for a few months, as if we don't occupy it, it will get redistributed to another institute and we will then never get it back".
Same might go for teaching assignments, e.g. a math chair teaching basic calculus to engineers, economics,... might be a source of income fought for.

These are of course only examples and there might be cases where saving it is possible (so feel free to wait for more answers), but from these experiences I am fairly certain that using the money is the better choice.