Herbs used to help with hyperthyroid symptoms include:
Bugleweed
Lemon balm
Motherwort
Gromwell
Rosemary
Sage
Herbs to help with hypothyroid symptoms include:
Gotu Kola
Ashwagandha
Coleus Forskohlii
Herb used for both hyperthyroidism and hypothyroidism:
Bladder wrack
I just did a quick search and found an article that contained a list of herbs one should avoid with hypothyroidism:
Astragalus - can contain potentially toxic levels of selenium and can interfere with medications that suppress the immune system, blood sugar and blood pressure--which may be fatal.
Bladderwrack - Prolonged use of bladderwrack has been shown to cause goiter and increase the risk of getting thyroid cancer. Bladderwrack, being a seaweed, is also prone to high levels of toxins such as heavy metals like arsenic found in ocean water.
Echinacea - many thyroid conditions and cancers can worsen with the use of antioxidants
Garlic - one of the adverse reactions of garlic intake is the onset of hypothyroidism. (my note: this refers to taking garlic as a daily supplement; it doesn't mean you can't use garlic as an ingredient in cooking occasionally).
https://www.livestrong.com/article/440613-herbs-to-avoid-with-hypothyroidism/
You should also be very careful taking supplements that contain iodine, such as Kelp or other seaweed products, if you have Hashimoto's as iodine can make autoimmune attacks greater.