I have Belarus citizenship but don't have a valid Belarus passport. Instead, I have US asylee travel document which indicates my Belarus citizenship. Belarus citizens don't need a visa to go to Brazil and Russia. Do I need a visa with a US travel document and no passport?
I don't know about Brazil.
I discourage your visit to Russia if you are a refugee from Belarus. As there is minimal border control between Russia and Belarus, the USA may believe you have visited Belarus. This may lead to cancellation of your refugee status.
The visa exemptions that you mentioned apply when you present a Belarus passport.
But, it seems that you are not able to obtain a Belarus passport, and are going to travel on a US refugee travel document. This document shows your place of birth, but it does not provide evidence of your nationality. It does not prove that you are a citizen of Belarus. Only a document issued by Belarus can do that.
In the case that you travel on a refugee travel document, both Brazil and Russia will require a visa in advance.
You also should remember that you may lose your asylum status in the US if you return to the country from which you fled. You may be asked about your travel when you return to the US, if the CBP officer believes you may have traveled to that country.
Regarding Brazil, sadly I could not find any general guidance on entry regulations for refugee travel document holders. Nonetheless, I did found an entry in the Brazilian embassy in Berlim (Germany) which is generic enough and shouldapply to your case.
3) Entry regulations for Refugees, stateless persons, holders of a "travel documents"
Refugees, stateless persons or persons with undetermined citizenship and holders of a travel document, that was issued by a country whose nationality they DON’T have, (e.g. German "Passersatz" or "travel document for foreigners") always need a VISA to enter Brazil.
In your specific case, you were born in Belarus, but do not have a Belarussian passport and also you have a refugee travel document, issued by the USA, but you are not a citizen of this country. According to the entry I mentioned earlier, you would need a VISA to enter Brazil.
All the information you might need to know about Brazilian Visas, like which one do you need, where and how to ask for it you can find in this page from the Ministry of Foreign Affairs.
Hope you have a nice trip to Brazil!