I don't use separate burners for milk and meat during the year, nor do I think there's any reason to do that. So why the need to kasher burners before passover?
The requirement to kasher burners is mentioned by RMA in Shulchan Aruch (451:4).
Mishna Brura (451:34) states that it isn't really necessary, since food absorbed by one utensil cannot pass to a second utensil just by touching one another (there needs to be hot liquid involved). Furthermore, even if some food spilled on to the burner, it likely burned up completely. However, libun kal is still needed due to the severity of the chametz prohibition.
You'll find that many laws of kashering apply only on Pesach for the same reason; i.e. we are more stringent with the Chametz prohibition.
Adding to Barry's answer:
Year-round, this isn't so clear-cut. I believe Chabad-Lubavitch's practice is to have separate burners. Rabbi Hershel Shachter (mp3 link) quotes his mentor as telling students, "when you get wealthier, you should buy separate stoves."
Even for Passover, I looked at the Chicago Rabbinical Council's guidelines, and it doesn't appear that they require kashering the grates, just cleaning them and leaving them in-place while the flame is on for a while. But it appears that Baltimore's Star-K does.
We have a general tendency to be stricter with Passover than meat-and-milk; here are several reasons why:
- It's just a few days, vs. all year round, so it's less hardship on people.
- איסורא בלע -- Chametz is treated as non-kosher, whereas meat and milk have certain leniencies because each are kosher on their own. E.g. if a potato is boiled in a clean "meat" pot, Sephardic Jews would allow it to be eaten with milk. Boil that potato in a clean chametz pot instead, and it's prohibited on Passover.
- עונשו כרת -- The prohibition against Chametz carries a more weighty penalty, "spiritual excision."
- באלף לא בטל -- unlike normal kosher guidelines, minute quantities of Chametz aren't "nullified."
- לא בדיל איניש מיניה -- People could space out and forget that it's Passover, as they're used to eating kosher Chametz products all year round. Whereas if you woke up the average kosher-keeping Jew at 2AM, s/he'd probably still remember not to eat cheeseburgers.