What is the procedure when being vectored for an approach and ATC allows you to pass through the final approach course?
Usually when ATC has a need to vector you across the final approach course, they will tell you about it before they do.  Something along the lines of "N1234 fly heading 230, vectors across final for spacing.".  If they don't and you see that you are getting close, you should ask them if they want you to intercept the course.

The AIM addresses this in paragraph 5-4-3 Approach Control:


  (b) After release to approach control, aircraft are vectored to the
  final approach course (ILS, MLS, VOR, ADF, etc.). Radar vectors and
  altitude or flight levels will be issued as required for spacing and
  separating aircraft. Therefore, pilots must not deviate from the
  headings issued by approach control. Aircraft will normally be
  informed when it is necessary to vector across the final approach
  course for spacing or other reasons. If approach course crossing is
  imminent and the pilot has not been informed that the aircraft will be
  vectored across the final approach course, the pilot should query the
  controller.
  
  (c) The pilot is not expected to turn inbound on the final approach
  course unless an approach clearance has been issued. This clearance
  will normally be issued with the final vector for interception of the
  final approach course, and the vector will be such as to enable the
  pilot to establish the aircraft on the final approach course prior to
  reaching the final approach fix.


The FAA Air Traffic Control Order Section 9 - Radar Arrivals also has the following directive for approach controllers:


  5-9-3. VECTORS ACROSS FINAL APPROACH COURSE
  
  Inform the aircraft whenever a vector will take it across the final
  approach course and state the reason for such action.
  
  NOTE-
  In the event you are unable to so inform the aircraft, the pilot is not expected to turn inbound on the final approach course unless
  approach clearance has been issued.
  
  PHRASEOLOGY-
  EXPECT VECTORS ACROSS FINAL FOR (purpose).  
  
  EXAMPLE-
  “EXPECT VECTORS ACROSS FINAL FOR SPACING.”