I have heard that agnosticism seems to be the only position with respect to god that doesn’t have a burden of proof. What I find troubling about this is most people do not as a practical matter think they have no knowledge on whether 10,000 mermaids are surrounding them right now. They simply don’t believe it.
There, you seem to be using "agnosticism" in the sense of having no belief either for or against the existence of God (or of gods, if you prefer). That is the definition I would use, too. This position does not require proof and is not open to disproof because it does not assert any proposition. In that sense, it has no burden of proof. The same is not true of any theistic belief. I would say it is not true of atheism, either, but some who claim to be atheist take positions that I would characterize as agnostic, myself.
Why isn’t this the case for god?
I don't believe I am surrounded by 10,000 mermaids because I think that if I were, I would perceive them. The same is not necessarily true of God. A person can rationally refuse to take a position on the existence of God based in part on a belief that they would not perceive God whether He exists or not. This is in line the ideas of gods that many people hold.
A further problem is in this. Let us suppose that there are many nonexistent entities. Presumably, there are an infinite number of them.
I guess by "there are" you mean something along the lines of "we can imagine".
Isn’t it more rational to simply assume that an entity without evidence doesn’t exist rather than give them a small chance of existing.
This has nothing in particular to do with agnosticism. Assuming is not the same has holding a belief, and assuming something does not exist is not inconsistent with allowing a small chance that it actually does exist.
For starters, it is impossible to distinguish between an entity with no evidence and a nonexistent one.
Well yes, I would take that to follow fairly directly from the definition of "evidence". But what does that have to do with belief? You seem to be working from an unspoken premise along the lines of "we should disbelieve that for which we have no evidence", but that is not sound.
Consider: I have no evidence of extraterrestrial life. Is it then rational for me to affirmatively believe that there is no life in the universe other than that which originated from the Earth? No. Extraterrestrial life might or might not exist, and I submit that the most rational belief to hold on the question is no belief either way.
Now, if we were to instead tweak our rule to "we should not believe that for which we have no evidence" then what problem does that present for agnosticism? The agnostic does not believe in God. Neither do they disbelieve, but that's irrelevant to the rule.
In practice, of course, people do not necessarily accept even the revised rule, and I think you would have trouble justifying it other than as an axiom (i.e. something you believe without evidence).
Secondly, this wouldn’t necessarily make you dogmatic. You can simply assume X doesn’t exist but change your mind if evidence came forth for X.
I have no idea how dogmatism comes into the picture, but again, assumption is not the same as belief. In practice, I think most agnostics do live day to day as if God does not exist, which is something like your "assume [God] doesn't exist". I think many would be prepared to consider evidence of God if it were presented to them. And those who would reject such evidence without consideration would be better characterized as atheist. So again, where's the problem with agnosticism?