I have access to lots of fresh horse manure, and would like to use it to start a hot bed. However, it contains sawdust, so would that prevent it from heating up enough to do any good?
I wouldn't put fresh horse manure directly into the garden, unless you were going to let it age for a good long time (like over Winter, perhaps) before planting anything in it. The fresh manure will be too "hot" and could damage your plants.
However, if you're going to compost the manure, which I think you really should, the sawdust is probably an excellent addition. It will be a "brown" to offset the "green" of the manure, and give you a better composting mix.
What do you mean by hot bed? Are you hoping to heat something like a cold frame with the material?
I have considerable experience using horse manure mixed with sawdust bedding. Most of it I have composted for a year or two before incorporation although I have been developing a large vegetable garden and have incorporated massive amounts. Much of the heating potential depends upon the relative amounts of sawdust and manure/urine that are combined.
A hot bed, or hot frame, is a cold frame with a heat source added. There once was a lot of information available about using manure to make a hot bed but I'm not finding it now, most are using electric cables today. A manure based hot bed needs to be made a bit ahead of the time of intended use to give the manure time to adjust to the heating needs, but basically yuou dig a trench, fill it with manure and a Carbon source (sawdust is fine), wet the material, and then put plants in the frame when the soil temperature stabilizes.