My big toe is partially numb after my sesamoidectomy 3 weeks ago. Will the numbness go away? How long does this take?

Probably, months. Depending on what sesamoid was taken out, some nerves can get stretched or damaged. A stretched nerve will usually recover over a few months but improvement may continue for up to 18 months. A damaged nerve may recover, bit there may be some permanent numbness. Only time will tell and there is not much to do in the meantime.
Numbness. Surgery may cause tissue irritation (expected). Tissue healing involves the skin, soft tissue , and surrounding nerves. It could take 9 months for healing including the numbness (neuroma). If you can move you toes, then nerve us not damaged. Massage area it helps healing, give it time.