How are Landing Fees Collected?
This depends on the airport but at KPNE where I used to fly there was a $5 landing fee (paid to the city of Philadelphia) that was collected at the FBO (where I rented the plane from) when I paid for the rental. I think if you do not pass through an FBO they will hunt the tail number down and send a bill to the party that has the aircraft registered. Its tough to fly into an airport and not pass through the FBO (unless you are just doing touch and goes). I would assume that for commercial traffic the airline/owning body pays and may even be billed by the airport monthly but I don't know the specifics I only know what I have seen for GA traffic.
If you are curious you can check out the Philadelphia Airport (KPHL and KPNE) legislation here which outlines the various fess and collection mechanisms the city imparts. Keep in mind these are city owned airports, see section V for details on collection but in short,

Section V Part d
Non-tenant Air Carriers and Operators not providing adequate surety or security in form and amount satisfactory to the Department of Commerce, to ensure payment of fees and charges, shall be required to pay all incurred fees and charges prior to aircraft departure.

For the point of contrast lets take a small privately owned airport here in the US, Solberg Airport in NJ has &quot;ramp fees&quot; for parking and the such, but like most small airports like this, these fees are waved with the purchase of fuel (usually at least some minimum amount). There would be no way to avoid paying (well Im sure there is) since usually the only way off the tarmac is through the FBO.