I would initially say all error messages that indicate a resource shortage (eg, the 1105/1204/701 you already mention) should go to the errorlog.
Of course, once you start dumping the extra messages to the errorlog, someone's going to ask for a method of disabling them (eg, may not care too much in a development environment; would prefer to choose which to log). So probably a bit more than what you're looking for ... but a means of flagging specific errors as (not) dumpable to the errorlog would probably be nice (something along the lines of how a repserver error class can be modified to ignore/warn/retry/log individual error numbers).
Then taking it one step further, allow user-defined error messages to be dumped to the dataserver errorlog, too.
Of course, then another question would be what to dump along with the error #/message, eg, login id? application? ip addr? In other words some sort of context would likely be of use, eg, allow the DBA to track the error back to its source.