This is a subject of controversy here. A straight T4 med may not show a falsely elevated thyroid test. Any T3 will.
As far as medical reverence I have not seen any, but I know from personal experience that dessicated will show an elevated T3 spike if taken before testing. Direct T3 will raise FT3 levels for 4-5 hours 20 minutes after ingesting. This can make your TSH temporarily drop. But FT4 will not, in most people go to a falsely elevated level.
So if you are still a little hypo and might need more meds, the doc will be reluctant to do so per the temporarily elevated levels of T3. Many doctors get freeked out from high T3 levels. If you have known heart issues than this might be a concern.
Getting thyroid tests done at early AM is best. You can then take ERFA sublingually after the appointment even if there is food in your stomach as long as there is no food remains in your mouth that could bind with the hormone. Most docs will allow you to schedule a lab time separate of the doc appointment time with only one visit fee.
The advice given by most sites I have looked at, state don't take your medication till after the testing. Try to aim for the same time of day, without taking medication.
Every doc I have spoken to says the same thing. Don't take it till after the test, do pick the same time each test, (thyroid levels fluctuate over the course of the day, not dramatically but enough) and be consistent in that!