Why does my bicycle steer funny and feel unstable when I have a heavy load in my Bob Yak trailer?

I suspect Chris H is on the right track with his weight distribution question.
You may have too little weight on the front wheel. An easy way to test too little weight on the front wheel is by shifting your body weight forward when the bike feels unstable.
Experiment with the trailer load to find the best handling.
You may find that adding front panniers with some weight helps.
It's all about experimenting until you find the right distribution.

Looking for other people who are having problems with stability using a Bob Yak trailer has led me to some advice that might help.

In a thread on campfirecycling.com Jacob says:

I’ve been a bike messenger (in The Netherlands) for some time, using a Bob Yak professionally. Since my positive experiences with this handy one-wheeled-trailer I own one myself now, mainly for use as a vacation trailer.

When loading a Bob Yak/Ibex trailer you should keep a few things in mind. 1. Always distibute the load evenly along its length-axis 2. place the heaviest load nearest to the trailer’s wheel 3. When you overload a Bob trailer (which is fairly possible), never make the load ‘top heavy’.

I’ ve handled loads of over 50 kilograms with the Bob Yak and have had no stability or handling problems with it.

Tip: A bike without shockabsorbtion is most ideal (in terms of handling/control) for towing a one-wheeled-trailer like the Bob Yak.

Another problem some people had with the Bob Yak is high speed stability wobbles. cara2u posted this warning on forums.adventurecycling.org:

Please be very careful if you decide on a one wheel Bob. If they are not loaded correctly they can cause your bike to completely lose control at the most inopportune time, downhill.
On our last TA tour we met an Aussie gal that had that very thing happen to her on a downhill. She was unconscious for two hours down in a drainage ditch until someone found her. She had to fly home to convalesce for two months before she could continue her tour. She was East to West and the accident happened the first week or so of her trip.

There is a lot more conversation about stability at speed in this thread.