Paragraph 2-1-2 of the FAA's Air Traffic Control Order (JO 7110.65z) states the following (pertinent excerpt):
2−1−2. DUTY PRIORITY
a. Give first priority to separating aircraft and
issuing safety alerts as required in this order. Good
judgment must be used in prioritizing all other
provisions of this order based on the requirements of
the situation at hand.
A NOTE attached to paragraph 2-1-2 states (in pertinent part):
... Each set of circumstances must be evaluated on its own
merit, and when more than one action is required,
controllers must exercise their best judgment based on the
facts and circumstances known to them. ...
Also, paragraph 2-1-4 of JO 7110.65z states (in pertinent part):
2−1−4. OPERATIONAL PRIORITY
It is recognized that traffic flow may affect the
controller’s ability to provide priority handling.
However, without compromising safety, good
judgment must be used in each situation to facilitate
the most expeditious movement of priority aircraft.
Provide air traffic control service to aircraft on a “first
come, first served” basis as circumstances permit,
except the following:
(emphasis is mine)
As noted in the paragraph above there are exceptions to the "first come, first served" operational priority. For example, AIR EVAC/MEDEVAC flights, emergencies, etc. would have priority over IFR or VFR aircraft that are not experiencing circumstances requiring higher priority handling.
So, if you are, as in your question, operating IFR, ATC should allow your approach on a "first come, first served" basis unless "Duty" or "Operational" priority require otherwise. Also, unless there are some unusual or special circumstances, ATC does not have the procedural authority to arbitrarily deny you an approach and force you to land at an airport other than the one you have been cleared to in your IFR clearance.
However, you should note that some very busy airports may disallow touch and go operations, low approaches and practice instrument approaches. (for example, see the "Additional Remarks" section regarding Hartsfield - Jackson Atlanta International Airport KATL - source: AirNav.com).