Okay, now that we've had some clarification, I will attempt again but more or less disregarding the homozygous/heterozygous parts.
Homozygous black + Heterozygous chestnut ( Chestnut + palomino) = ________? If your black is actually homozygous black, your options are black, bay, or brown. The agouti gene mentioned above that restricts black into bay, brown, or wild bay can be carried by chestnuts but you wouldn't know without testing because chestnuts are homozygous red and have no black to restrict.
If you have a heterozygous black + chestnut, you'll get black, bay, brown, or chestnut.
Heterozygous grey ( Gray and bay)+ Heterozygous red dun ( Dun + Chestnut)= ________? The grey would be heterozygous here because it only has one gray parent. However, I do not know the grey horse's base color (as every grey is a "normal" color underneath, but grey covers it). So I can't answer that, but I can tell you it will have a 50% of being grey.
Homozygous bay+ __________ = Palomino? Well, the bay couldn't be homozygous black or palomino wouldn't be possible as it is red-based. It would also have to be something with cream and heterozygous black, so smoky black, smoky cream, buckskin, perlino, palomino, or cremello.
Homozygous chestnut tobiano+ ________ = Heterozygous bay overo? I'll consider the tobiano heterozygous even though I think you mentioned that you meant to put tobiano instead of overo. I'll also just go with the word overo rather than the individual patterns. Your color options would be really any black based color that is heterozygous for all of its other genes (ie, buckskin is heterozygous for cream, a blue roan that only has one copy of Roan, a dun with only one copy of dun, et cetera) plus one or more overo genes.