It’s reasonable to be concerned that your hamburger may once have raced in the Grand National, but worries about chemical contamination from the horse meat are almost certainly overblown. In fact, your health might be more at risk from the burger itself:
There is reasonable public outrage at possible criminal conspiracies to adulterate meat products with horsemeat, and additional concerns raised about the presence of the anti-inflammatory known as bute.
While not in any way questioning this concern about adulteration with a chemical compound, it is helpful to get a sense of magnitude. When bute was given as a human medicine, it was reported to be associated with a serious adverse reaction in 1 in 30,000 (over a whole course of treatment), but at a dose giving concentrations at least 4,000 times that arising from eating a diet of horse meat – see the excellent information from the Science Media Centre
So making all sorts of heroic assumptions about there being a linear-no-threshold response, we might very roughly assign a pro-rata risk of a serious event as 1 in 100,000,000 per burger.