Short answer: no.
"Then why do I see a comma after but so much?" you ask.
"Because people are wrong," we answer. "And also because there's an exception."
Parenthetical elements are set off from the rest of the sentence with commas, like this:
We baked a cake in the shape of a spider for the Halloween party. But, regardless of our time and dedication, the arachnid looked more like a misshapen black blob.
That side comment (regardless of our time and dedication) isn't an essential piece of the sentence puzzle, so there are commas around it. Because it comes right after but, the commas is… right after but.
But if there were not parenthetical information, there'd be no comma. Then it would look like this:
We baked a cake in the shape of a spider for the Halloween party. But the arachnid looked more like a misshapen black blob.
No parenthetical information; no comma.
P.S. Don't get confused and completely eliminate the commas near but. When but is used to connect two independent clauses, you need it—before the but. Like this:
We baked a cake in the shape of a spider for the Halloween party, but the arachnid looked more like a misshapen black blob.