Should I run a relay continuously?
Look for a high quality relay that has a rated maximum continuous coil voltage. As long as the voltage driving the coil stays below this value and the environment (temp, humidity, etc) stays within the rated ranges then you should expect to be able to drive the coil continuously for its rated lifetime. 

One failure mode to consider is that relay contacts can become "stuck" shut. In this case, the relay does not open even though the coil is deenergized. This happens usually because of sparking across the contacts when they are connected to an inductive or high voltage load when they open. It can also happen if you over heat the contacts by driving too much current though them and getting them hot. 

It might be a good idea to periodically deenergize the relay both to exercise and to test it. Ideally, you'd do with without a connected load to minimize wear the contacts.