What's the best way to get rid of a nest?

Don't get me wrong, I'm asking this out of concern for the birds.

A couple of pigeons started cramming sticks under a tool cabinet on the balcony. The thing is, said balcony is VERY small and the cabinet is actually often used since many daily house maintenance tools are stored in there. The only reason why they've been able to start building up is that it was left untouched for some days.

Last time I checked there were still no eggs, but if they actually get to lay two and they hatch there's a high risk that they might end up abandoning the squabs since the area is often trafficked.

If i were to dispose of the sticks before any egg is laid, would that cause health problems to the pigeon? Or is it just going to lay them somewhere else?