Is it possible to determine the slit a photon went through in the double slit experiment by measuring its flight time? In the double-slit experiment, quantum mechanics states that if you try to determine which slit the photon goes through, you won't have a resulting wave pattern.
But, knowing the time it took for the photon to go from the source to the observing screen, can you deduct the distance of the photon path and so which slit it passes through (except if the photon impacts the exact middle of the observing screen)?
 A: 
But knowing the time it took for the photon to go from the source to the observing screen, you can deduct the distance of the photon path and so which slit it passes through

... and indeed such information will make it impossible for fringes to appear. 
Interference experiments use wavepackets that have a long duration, which makes it impossible to tell from timing information which slit the particle came through, eliminating the problem.
A: There exist single photon at a time  experiments though for a double slit experiment the single electron at a time are easier.
That the interference pattern disappears if one places detectors to check the path is an experimental fact, it is the result of the mathematics of quantum mechanics which up to now really describes the microworld of particles and atoms.
As it is clearer for electrons, I will start with them:
a) There exists a wavefunction a solution of a QM wave equation, which may display wave characteristics of interference given the correct boundary conditions , which pick the solution.
b)The double slit experiment is a boundary value problem for the quantum mechanical regime. It is the boundary problem "electrons of specific momentum scattering off two slits of specific size".
Nature gives the solution as an interference pattern. 
If one puts a detector on the path of the electron, the boundary problem changes to " electrons of specific  momentum  scattering off two slits of specific size and rescattering off detector on the way". Let us suppose that the detector at the screen is an active pixel with good time resolution so only a detector on the way is needed.
What happens is that any detector on the way will scatter the electron, i.e., a different wave function will be describing the setup.The interference pattern is not seen because the secondary scatter spoils the phase of the solution, with respect to the geometry and no interference appears.
That it is the change in the boundary conditions that destroys the phases and thus the interference pattern  was made clear in a recent photon experiment, where they show that by trying to locate which way the photon went, the photon reaching the detector loses the phase and leaves the interference pattern.
The same would happen with any detector, either positional or timing detector. The change in the boundary conditions that the detector introduces destroys the original phase correlations that gave the original interference pattern. 
A: Just try to think of a thought experiment by which you will measure the time of travel for the photon:
You will need to start a timer when the photon gets emitted by the source. Thus you'll need to Measure its position once.
Then you'll need to measure it when it reaches the screen to stop the timer.
But here's the catch: photons are indistinguishable. So just by measuring at the start and at end you cannot say it was the same photon. So you'll need to trace it throughout the way. But everytime you measure you are disturbing the system. Thus measuring the photon even with very low energy rays or fields you'll disrupt the process of interference and there will be no fringe pattern.
P.S. Feynmann has a brilliant way of explaining the whole business. Go through Feynman's lectures in Physics, volume 3. It's really entertaining.
