If you are on a heading, being vectored to intercept final on an instrument approach, and it appears that you will fly through the final approach without being cleared to intercept it, what should you do?
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.
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.
EXPECT VECTORS ACROSS FINAL FOR (purpose).
“EXPECT VECTORS ACROSS FINAL FOR SPACING.”
This is one of the most confusing areas of IFR flight. Controllers will use many different phrases like "turn to heading XYZ... intercept the localizer" or "... join the localizer" or "... report established" etc. Oftentimes before they will issue the clearance, which they hold back as long as possible. They want you to turn to XYZ and then turn again onto the final approach course. This would appear to be against their guidance but it happens quite often. Note, you are not "cleared" for the approach yet, even though you are maneuvering onto the final approach course. You must maintain your last assigned altitude and speeds until cleared. To answer your question, if you are "turned to heading XYZ" but not cleared for the approach nor told to "join" the final approach course/localizer/etc., I would confirm if the controller wants you to fly through the final course (or he just forgot about you).
In addition to the other answers, there is one more thing to consider - lost comms.
This scenario here is a classic for thinking about what your actions will need to be in the event of loss of communications.
Keep track of WHY those vectors are being given to you - where are they taking you, when should you expect headings that would make sense in regards to shooting the approach, etc. Always be questioning what ATC tells you to do (and of course call them if it does not make sense). Always try to think "what comes next", this will help you prevent problems when "what comes next" is an error on either their behalf, or yours!