In this context, to tap (someone) does not mean physically tapping them.
Historically, tapping someone on the shoulder is an action done to select someone or to designate someone, either to assign membership in an organization or to delegate a task of some sort.
The simplest example of this would be childhood team sports, where the “leaders” of the two opposing teams for a class sport (example: soccer, or what most nations call football) take turns selecting members of their school class for their own team. This would be tapping the children to be players for one side or the other. This could include a physical tap, though a finger point and calling out someone’s name works just fine (and saves time) in such an informal situation.
So, when President Obama taps Geithner to aid with the tax debate on Capitol Hill, Obama is not appointing Geithner to anything; Geithner is already a Senate-confirmed member of Obama’s presidential cabinet. He is, in effect, a minister. Rather, Obama is designating Geithner to assist with this specific task, even though it is outside Geithner’s normal duties, being essentially an effort to politically lobby members of Congress to take Obama’s side.
It is in this sense that Obama is tapping Treasury Secretary Timothy Geithner and Office of Management and Budget Director Jacob Lew.