Feynman diagrams are drawn on Space-time graphs. So is it possible for any particle path in that diagram to have a slope smaller than 45° or greater than 135°, i.e. can they travel faster than light? Does the slope or or velocity of the particles have any effect on the outcome of the diagram? Virtual particles (photons and the like) should have no limits for their properties, but what about incoming particles, say electrons?
Incoming/outgoing particles with mass must travel at less than the speed of light. Incoming/outgoing massless particles must travel at the speed of light. The “virtual particles” inside the diagram can “travel” at any speed, but since they don’t really behave like particles this doesn’t matter.
Feynman diagrams aren’t pictures of what is “really happening”; they are just calculational tools. “Virtual particle” is an unfortunate misnomer, and the emphasis should definitely be on “virtual” rather than “particle”.
Feynman diagrams are not drawn on space time, that is a common misconception. It helps to think of them as being drawn on space time, but the reality is a bit different.
Feynman diagrams are nothing more than abstract diagrammatic pictures representing various terms in the expression for the transition amplitude between quantum states.
Feynman diagrams are multi-faceted, since their origin . Feynman starts by transforming equation of motion (Schroedinger's or Dirac's) into an integral equation and draws diagrams according to the title, i.e. in spacetime. He names $K_+$ the kernel (later known as "Feynman's propagator").
Sec. 6 of the paper is entitled "Energy-momentum representation" and here you may find the form of Feynman diagrams in general use afterwards.
In path integral formulation the integral is over configuration space for particles, over field configuration space for fields. In both cases, not over spacetime.
 R.P. Feynman: "Space-Time Approach to Quantum Electrodynamics", Phys. Rev: 76 (1949), 769.