If you continuously send UDP messages to a server that isn't accepting them, you will eventually get an error from the sendto() system call; the uninterested receiving host is generating ICMP messages saying "I don't want these". It's true that there is a flavor of sloppy socket coding where that error doesn't manifest itself. But if you're writing good code, TCP is no less fire-and-forget than UDP.

UDP is marginally faster than TCP, but the tradeoff for that is that under heavy load, UDP imposes more costs on the rest of your traffic. Since we're talking about logging, though: who gives a shit how fast it is? With either transport, if logs are taking more than hundreds of milliseconds to clear, you have a problem you need to fix.