How to determine if two mutations in a gene are on the same or different alleles?
Yes, it can be determined by sequencing but the sequencing read has to be long enough to span both the mutations. PacBio sequencing allows long reads but it also has higher error rate than the short read sequencing platforms like Illumina. If you know the gene locus then you can amplify the region using PCR and use Sanger sequencing too (for this the amplicon has to be cloned in a plasmid, transformed into bacteria and then single colonies must be sequenced; see Why does one combine PCR and cloning as ways for amplification of sequences?). 

Another possibility is to look at the RNAs from this gene. If the two alleles have different expression then comparing the number of reads for each mutation might tell if they occur together. However, as I said it already depends on the two conditions: expression and dissimilar expression.