Levothyroxine is generally very well tolerated. Some people are actually allergic to the dye in levothyroxine, but there is definitely a work-around using a [dye-free] 50mcg pill. Other symptoms can include nausea and vomiting.
Another side effect that can happen, especially whenever you're first starting the dose, is fatigue and tiredness. Within the first few weeks of taking levothyroxine, some people can experience change in their menstrual periods, diarrhea, insomnia, nervousness, and anxiety.