Why is a clock necessary for IFR?
There are several reasons:


For departure clearances. An IFR clearance may have a void or release time so that clearance is only valid between certain times.
For holding pattern legs. A standard holding pattern is based on a time of one minute at or below 14,000' MSL and 1.5 minutes above 14,000' MSL on the inbound leg.
For holding pattern clearances. Holding instructions include an EFC (expect further clearance) time, so that if you have a communication failure while holding then you know when you can safely leave the hold and start an approach.
For timed approaches, where an aircraft is cleared to start an approach at a specific time
For timed turns. A standard rate turn is 360 degrees in 2 minutes, so with only a timer and a turn coordinator you can turn a specific number of degrees.
For en route reporting and general time/distance calculations. Especially in a non-radar environment (i.e. when ATC can't see you on radar) you may be required to give ATC actual or estimated times for passing certain fixes. (This is not strictly IFR only, but it's still relevant.)


All of these scenarios are discussed in detail in the FAA's Instrument Flying Handbook and in the AIM.