When are pilots required to have approval to conduct practice approaches?
Usually airports with approach controls will have it printed on the approach plate. Take for example the non-towered field Menominee-Marinette Twin County:



The approach control frequency is highlighted in the chart. This is who you receive approach clearance from. KMNM is outside of KGRB (Green Bay) airspace by a good 40 miles, but KGRB approach control is responsible for the airport. 

So the short answer is: Look at the approach plate for the responsible frequency/agency for the airport. 

I think the other part of your question is more along the lines of "If I'm shooting an IFR approach, am I required to get clearance via ATC?" This I'm not 100% sure about but I would guess that the answer is "yes" since you don't want to interfere with traffic shooting real approaches. 

The Aeronautical Information Manual has advice for conducting practice approaches in section 4-3-21 (Page 204):


  (a) ...Practice
  instrument approaches are considered to be instrument approaches made by either a VFR aircraft not on an IFR flight plan or an aircraft on an IFR flight plan.
  To achieve this and thereby enhance air safety, it is Air Traffic’s policy to provide for separation of such operations at locations where approach control
  facilities are located and, as resources permit, at certain other locations served by ARTCCs or parent approach control facilities.


(Emphasis mine)

Furthermore, it goes on to say...


  (b) Before practicing an instrument approach, pilots should inform the approach control facility or the tower of the type of practice approach they desire to make and how they intend to terminate it, ... 
  (c) At airports without a tower, pilots wishing to make practice instrument approaches should notify the facility having control jurisdiction of the desired
  approach as indicated on the approach chart. 
  (e) VFR aircraft practicing instrument approaches are not automatically authorized to execute the missed approach procedure. 


But as I say in my comment, the AIM is a guideline, not a rule and I don't think there is anybody stopping somebody from doing an instrument approach without controller notification, however because ATC isn't providing traffic separation at that point, it could be considered negligent of the pilot to not request ATC clearance. Besides, a good portion of conducting practice approaches is talking to ATC.