Ebola in the White House?

What’s to stop people from spreading ebola and other biological agents during tours of the White House, accidentally or deliberately? Is there some sort of filtration system that separates the VIPs from the plebs?

Yes.

The full details aren’t known to the public (Cecil Adams always excepted), but the WH is over pressured, to force gas & pathogens out.

It is certainly filled with hot air, under high pressure.

Are the people in charge of security advised to be on the alert for sick tourists? As I understand it, someone infected with Ebola is only contagious if he is showing symptoms. It seems to me that the security people would be instructed to turn anybody away who is coughing, has the shakes or other signs of illness and might even hold him for pickup by emergency medical personnel if he is showing severe signs of illness

I suspect you’re right. Let’s hope they’re particularly alert these days.

An episode of The West Wing showed a very sophisticated CDC-quality small clinic at the White House. That’s Hollywood; I doubt the facilities are that good. For more: Physician to the President - Wikipedia