In BEAST2 this is a little bit fiddly for the moment. But basically you should just load your variant-only alignment into BEAUti as normal and produce an XML. Then you will need to hand-edit the XML Alignment element as follows:
<data spec="Alignment" statecount="2" weights="1517,141,1526,487,1,1,1,1,1,1,1,1,1...">
<sequence id="seq_C1" taxon="C1" totalcount="4">ACGT ACACA</sequence>
<sequence id="seq_C2" taxon="C2" totalcount="4">ACGT CACAC</sequence>
...
</data>
In the Alignment element you should add "ACGT" to the front of every sequence as shown above.
Then add the weights="X,Y, Z,W,1,1,1,1,1,1,…" attribute to the Alignment element as shown above.
X, W, Z, W are the counts of the number of invariant sites in the genome of each of the 4 nucleotides A, C, G, T, respectively. These weights should be followed by a 1 for each of the variant sites in your alignment.
We are working on a better way to do this in the future...
Cheers
Alexei