Does the phrase "but although" work together?

The sentence is OK. The words but and although are not the same, since if they were the same, either word could be left out and the sentence would still make sense, and have the same meaning:

• Omit but and the sentence is still correct and has the almost same meaning:

  "Although you hate me now, I believe that you will forgive me over time."

• Omit although, and there appears to be a word missing or a punctuation or syntax problem:

  "But you hate me now, I believe that you will forgive me over time."

  And yet notice how the meaning can be preserved by moving the leading conjunction to the middle:

  "You hate me now, but I believe that you will forgive me over time."

Tautology

This is a tautology. The definition of tautology according to OLD:

The saying of the same thing twice over in different words, generally considered to be a fault of style (e.g. they arrived one after the other in succession).

BBC Academy has a guide called "BBC News style guide", which includes a page on grammar, spelling and punctuation, it says, among other things, the following (this is a direct quote from their page):

Tautologies

Try to avoid them. Common examples include:

advance warning, armed gunmen, universal panacea, She has given birth to a baby boy, mutual co-operation, fixed phone line, local resident, crew members, past history, exact replica, anti-government rebel forces, pre-conditions, pre-planned, Sharia law (Sharia means Islamic religious law), weather conditions

Double contrast

In the cases illustrated above, it is quite clear what one means, for example "armed gunmen" are clearly people who are armed (the added armed does not call into question the meaning of the word gunmen).

In your case this is different, both but and although indicate contrast, so the but serves no purpose in "But although". Therefore, one could just write:

"Although you hate me now, I believe that you will forgive me over time."

When "but although" can be correct

The above does not mean the combination "but although" is always wrong, they can be used in a way where both contrast on different aspects. I do not have a proper example, but I can refer to this answer on ELL, which argues (also without an example of the two words being used next to each other) both words can be used in a contrastive conjuncture.