Fallacy of when an arguer attempts to minimize a fact?

That is a fallacy, or a simple error of fact if it suggests something factually inaccurate. The detail depends on the context:

"A mosquito is nothing to worry about because it is so small"
Well, it is small for a man to contemplate, but it can in fact be something to worry about if it could carry dengue, or yellow fever or malaria. But that in turn does not mean that because it is small, it must be deadly.

"A dog bite is nothing to worry about, because a dog is smaller than a man."

Similarly, ignore a huge dog biting a tiny man; a dog bite can cause a fatal infection, or rupture a major blood vessel, or tear out a man's throat. Sissy of course, but not nothing.

"A hippo bite is nothing to worry about because hippos are funny."

Simply counterfactual; a hippo can bite a man in two (usually a dumb tourist who got what he asked for), and on occasion has actually so.

Now, those examples are just examples. Which formal fallacies Bo would assign them to, I cannot say. They seem to resemble the No True Scotsman fallacy or dishonest argument (along the lines of "No true Scot would fall for that or do that" etc).

Or you could call it a simple non-sequitur: "Because it is possible for a small factor to be negligible, and there is no absolute criterion for smallness, therefore any factor can be dismissed as negligible."