Yes, your question makes alot of sense and its a good question because hits at the heart of differences between population genetic analysis and phylogenetic analysis. However, when you speak about gaps you are referring to 'indels' (insertion/deletion).
Indels The conservative approach is to simply delete sites with gaps. Arlequin uses a lot of maximum likelihood solutions and gaps would automatically be deleted in an Arlequin max. likelihood analysis, because it is the output per alignment per matrix. If the matrix doesn't have a 'gap'.
Indels and pop gen Thats tricky, it is better to remove them. HOWEVER the indel may contain underlying population signal, if its a maximum likelihood analysis (pop gen does this too) I suspect they will be ignored. If its pairwise, such Fst including indels might give you a better answer because there is more polymorphisms to assess population structure. So if you're doing a pairwise analysis, I'd do it both ways and then report the methodology correctly if there are differences in results and simply explain the theory.
Invariable sites This is different from an indel. An invariable site is a site which carries no phylogenetic signal, it carries information but a phylogenetic analysis could draw any conclusions because all the nucleotides/amino acids are identical at a give alignment position.
The output you obtain in allele frequency will be different and here's the rub ...
Phylogenetics analysis Arlequin is really pop gen, but it will contain some phylogenetics programs. If you see anything with 'Jukes Cantor correction', 'Kimura 2 (or 3) parameter model', 'HGW model', '"REV" model' (eek I can't remember its name any its a reverible 6-parameter model of nucleotides), you are dealing with phylogenetics. Keeping invariable sites is normal here because the rate matrix is calculation "per alignment site".
Population genetics analysis Fst, Fis, linkage disequilibrium statistics, REMOVE THEM. Why? Because you want SNPs (single nucleotide polymorphisms). The invariable sites will get in the way of the statistics. For example is the locus recombines and you want to assess a linkage disequilibrium statistic (recombination), invariable sites will recombine, but the algorithm can't see this because the nucleotides are all the same. The only thing the algorithm can see and analyse are SNPs, which 'representative' of all the surrounding invariant sites. Thus removing invariants is essential. Arelquin might do this automatically (but I don't know the preparatory algorithms), but if you submit multiple loci Arlquin will want to know which are different loci. It will then recode the loci automatically if a single locus comprises more than one SNP. We are going into the underlying theory of linkage disequilibrium here and I'm not sure if say Fst/Fis is particularly bothered with this or is just concerned with the populations of genetic diversity.
Summary So gaps ... if they are genuine indels and not just 'N', ie indeterminant nucleotide calls, remove them for phylogenetics. If its pop gen do the calculation both ways and assess the output, if there is no difference report the 'gap/indel removal' result.
Invariant sites .. for pop gen you remove them but Arlequin might do this automatically to prepare a data set for analysis (I don't know). For phylogenetics you never remove invariant sites.