Is it considered showing off or seeking a reputation for people to thank a Muslim for doing good, and is there any harm in giving zakat openly?

If people praise a good deed you have done, that is an immediate glad tiding, and it does not harm you unless you deliberately sought their praise and commendation and aspired to it, for that may lead you to showing off (riya'). The proof for this is the saying of the Messenger of Allah (peace and blessings be upon him) when he was asked about a man who performs a good deed and people praise him for it. He said: "That is the immediate glad tidings of the believer." This applies if people praise him without him seeking their praise; otherwise, deliberately seeking it is blameworthy.