Is it permissible for a fasting person to lie about their fasting by saying: "I am full, or sick, or my teeth hurt," if food is offered to them while they are a guest, with the intention of sincerity to Allah and concealing a good deed?

Concealing acts of obedience is religiously مطلوب (desirable), as it is closer to sincerity (ikhlas) and further from ostentation (riya'), unless there is a need to show them. This is supported by the hadith: "Whoever among you can have a hidden good deed, let him do it."

It is not meant to lie to conceal a good deed, but rather to conceal what is possible, even through subtle hints (ma'arid) rather than outright lies, just as Al-Hasan Al-Basri and Ayyub Al-Sakhtiyani did.

If you are invited while fasting, you can say, "I have no need for food now" or "I am fasting," as in the hadith, and this does not contradict sincerity. It is permissible to show acts of worship if there is a need for it, such as reconciling disputes (islah dhat al-bayn) and winning over hearts.