I found a handy clue that might help provide an answer to this. In A Leela Of Her Own, we meet Hank Aaron the 24th. Hank Aaron was born in 1934. Fry did the nasty in the pasty in 1947. Which means Fry's "grandfather" is roughly 10 years older than Hank Aaron, which is about half a generation. Because Fry is his own grandfather, I'm using Fry-> Yancy-> Yancy Jr. as the starting point here, as it simplifies the calculation of how much of Fry's DNA is in Yancy Jr's lineage.
If we were to assume that Fry's family had kids at roughly the same age as the Aaron's, then this means they both had 23 generations since then. Each of those generations means a halving of Fry's DNA. So to calculate how much of Fry's DNA is in the Professor, it's simply (1/2)^23.
However, there are ages of the end of those lineages to take into account. In the future present, Hank Aaron the 24th is retired, and his hair is graying. In modern Major League Baseball, most athletes retire around age 40, so we can assume Hank's at least that old, maybe in his 50s-60s. The Professor is 100 years older than him, which given modern generational lengths of 20-40 years means he's 3-5 generations older than Hank.
This means there would be 18-20 generations between Fry and The Professor. That means The Professor has between (1/2)^20 - (1/2)^18 of Fry's DNA. That works out to be 9.53674316 × 10-7 - 3.81469727 × 10-6.
After discussing in chat, a flaw was pointed out in this answer. We don't know that every descendent of Hank Aaron has his name. So the numbers above are a lower bound on the number of generations, and an upper bound on the amount of Fry's DNA in the Professor.