We have users connecting to Analysis Services via a web
app to browse cube data and run an incremental process
every week to update this data. We also run a full process
every now then as well.

Does anyone know if it is OK to allow them to browse cube
data while the Analysis Services database is being
processed(by both a full and an incremental)?

Any suggestions would be appreciated.

Thanks!

Yes,
You can allow users to browse data while cubes are being
processed. They need refresh the web page i.e they need
to reconnect to the cube to view new data.
Regards

When you process a cube, it will ask for full process ,
refresh and incremental update. Incremental update is a
better solution to process cubes just to save time.

while going for incremental update, u need to specify
filter like "year=1998" , AM will process the new data and
will merge it to the existing partitions.

you need to go for full process when any structural
changes has taken place for the cube. then user must
reconnect to cube.

Cheers
Yashwant