I agree with the previous responses, only I would add slightly stronger advice. The construction, "it seems not to be", is perfectly correct, and it is perfectly current too (i.e., not old fashioned). Nevertheless, when used outside of formal writing it sounds stilted or insistent.
Let me also give technical detail for anyone who enjoys it, and only for them. This pair of sentences from clapec is just one example of raising, a grammatical process. Because raising happens all over the place within English, while being used sparingly in most other languages, I think some readers will be glad to know more about it.
Raising applies to complex sentences (where one sentence contains another sentence) -- yes, getting very technical now. The following pairs of sentences show the process:
It seems not to be the case -> It doesn't seem to be the case.
It seemed (that) the noise got on his nerves -> The noise seemed to get on his nerves.
It seems that big hipped women turn him on the most -> Big hipped women seem to turn him on the most.
People consider that he is smart -> People consider him smart.
You "raise" a constituent like the noise from the "lower" (subordinate) clause to the "higher" (main) clause. Not originally modifies "to be", then it is made to modify "seem".
Unfortunately, this neutralizes a meaning difference. If raising in "seem" sentences were not part of the landscape of English grammar, then "it does not seem to be the case" would entail "but on the other hand, it doesn't necessarily seem not to be the case, either". Preserving subtle differences in meaning is one thing that really drives prescriptive grammarians when they impose rules of grammar. To try to increase the sophistication of a language -- well, that is a nice idea. But what's the reality? Languages do things in disregard to elegance of thought. English does apply raising, and ambiguity is generated.