How to use the infinitive in this sentence?
This is a situation where Neg-Raising is useful. You want the negative in the main clause:
- This does not appear to work any longer in any web browser.
Double any's in the sentence is fixable by Neg-Raising the whole phrase no longer
- This no longer appears to work in any web browser.
In English we far prefer to negate the main verb in the matrix clause in sentences like this. Negating the infinitive is generally awkward unless extreme technical specificity is what is required. Compare:
- I don't want to go.
and
- I want to not go.
- I want not to go.
The original is better because it negates the main verb and not the infinitive. It is easier to process.
The original Poster is using the phrase no longer to negate the sentence. They would be far better off with:
- This no longer appears to work in any web browser.
Here the negation is applied to the clause headed by the verb appears, not to the infinitival clause.
For the record, as it will no doubt be a matter of some debate here, there is no reason not to split an infinitive if it sounds right to you.
Your third one is correct:
This appears not to work any longer in any web browser.
You can also use a split infinitive, however some grammarians would advocate against it:
This appears to no longer work in any web browser.
Personally, I'd prefer the split infinitive version (sounds more idiomatic to me).
It appears this no longer works in any web browser
Why go through the tar pit when you can go around it.
However, another excellent question is, the word before "any," in or with?