Are there any restrictions on hire cars when you cross borders within the EU? I'm thinking about hiring a car in Linz, Austria and driving to Česky Krumlov, Czech Republic (about an hour away) and back again. I presume it's up to the rental company, but is it generally allowed?
It all depends on the conditions of the rental company, but especially travel to East European countries is often restricted or prohibited.
Just a few examples for your exact combination (rental in Austria, travel to Czech Republic): Sixt allows travel with the cheaper categories of cars, while Avis prohibits all travel to the Czech Republic.
There are three aspects to this question:
Border checks: while there are occasional spot checks on the Czechia-Austria border, they're mostly targeting illegal immigrants and smugglers of contraband. Immigration personnel certainly wouldn't care about the car you're driving, especially since you're not a local resident.
Police checks: it is of course possible that you would get stopped by the police for a random check, but again they don't care about the contract between you and the rental company. As long as you have your driving license and the car's documents, the police will be satisfied. @Bent mentioned that you might get in trouble for not having Austrian insurance, but that's not true:
When you register a car in any EU country, you must insure it for third party liability. This compulsory insurance is valid in all other EU countries.
Your contract with the rental company: while some companies do disallow driving outside the Czech Republic (although few would have issues with Austria), they pretty much cannot check that you haven't actually went there, unless there's a GPS sensor installed in the car. The only way they might find out is if you get into an accident outside the area where you've been allowed to drive and report it, so you'd better drive carefully while violating the contract.
My own advice is to find a company that would allow this trip (most would), rather than breaking your contract and facing the minor risk of getting caught.
Note that this answer applies to driving between any EU countries, not just Czech Republic and Austria.