Idea: Force politicians to leave office when they run for president or are selected vice president

[Del Tachi]

I believe this is actually a constitutional feature in many presidential systems.   But I'd probably want it to be worded like this:

"No Senator or Representative, or Person holding an Office of Trust or Profit under the United States, shall be eligible to the Office of President." 

[Del Tachi]

Why though? What exactly would be the benefit of it?

The rule protects an independent executive, as members of the legislature are required to give up their seats before they are eligible to run.  It also hampers a would-be autocrat interested in consolidating political power from inside the government, as he would be required to "sit out" of national politics for a requisite amount of time before assuming the presidency.  The rule is general protection against an oligarchy of career politicians or bureaucrats, which is something we actually pretty desperately need.