Are deadlocks a serious problem? I have a few production SQL Server instances and I get around 100 deadlock alert emails a day (up to 100 for one server, 150 in total).

Is this a problem I should investigate? How many deadlocks are problematic and what can be the possible consequence of ignoring this issue?
It's typically not the count of deadlocks that can cause a serious problem, but the reason to investigate would be to find out what workloads are deadlocking on each other.  After all, in a deadlock there is a victim.  So now you have a process that didn't complete the work that it intended to complete.  And, in good theory applications should have some sort of retry logic in them for situations like this... but in practice this almost never happens.

With that being said, investigate what is deadlocking.  Typically it is the same workload or type of workload that generates the majority of deadlocks.  A few quick fixes (after reading the deadlock XML) could be an easy win to get rid of most of them.

(yes, 100+ deadlocks a day is a lot.  But any recurring deadlocking pattern, whether it is 1 or 100, should be investigated)