Can you continue an ILS approach when weather falls below minimum?
You didn't list a jurisdiction so Ill answer for the FAA: Yes you can continue;

Under part 91 you can both start and fly an approach if the conditions are below minimum.

If you are flying under part 121 then you can't start the approach but § 121.651 Takeoff and landing weather minimums: IFR: All certificate holders. is going to be the regulation to reference and allows you to continue if you are passed the FAF. 


  (c) A pilot who has begun the final approach segment of an instrument
  approach procedure in accordance with paragraph (b) of this section,
  and after that receives a later weather report indicating
  below-minimum conditions, may continue the approach to DA/DH or MDA.
  Upon reaching DA/DH or at MDA, and at any time before the missed
  approach point, the pilot may continue the approach below DA/DH or MDA
  if either the requirements of § 91.176 of this chapter, or the
  following requirements are met: