I take levothyroxine every morning with a glass of water, and then wait at least 30 minutes before having any food or other drinks. I know taking supplements (or food/drink that contains a high amount) of calcium or iron can affect absorption, so I've cut out the milk I used to put on my oatmeal, and try to avoid vitamins/supplements in general until at least 4 hours later.
I frequently take a caffeine tablet about 1.5 hours after taking the thyroid med though, because I like to do morning runs and it helps with my endurance (I haven't been close to having hyperthyroidism, so maybe caffeine is frowned on by some, but my heart rate on the run seems fine, and I only take 100mg caffeine, so about a cup of coffee).
All that said, I'm also a "bad absorber" of levothyroxine which means I'm now taking a much higher dose than someone my weight should and still am not quite back to normal thyroid levels, but I don't think the caffeine is causing my problem.
I would think maybe the coffee could make things move faster throughout your digestive tract and might affect your absorption, but if it is something you enjoy, and if you have to take this thyroid medication for the rest of your life, find something that will work for you. As long as you are consistent, and wait about an hour before your coffee, you should be able to find the right dose for you and still have coffee.
Hope this helps - I'm not an expert and some people might come back and say caffeine definitely affects absorption, but all people are different and as long as you are consistent you should be able to find something that will work for you.
"The effect of espresso on L-T4 absorption was variable, and was present only if espresso was swallowed simultaneously with the L-T4, but not 60 minutes later."
"This pattern of taking espresso or coffee with L-T4 was highly consistent among the eight study patients."
Altered Intestinal Absorption of L-Thyroxine Caused by Coffee - Thyroid. Mar;18(3):293-301.