I wouldn't. Many Arts & Crafts houses were built in Tudor style with beams and other Tudor era features.
We've been fortunate to own two houses with Arts & Crafts attributes -
One was actually an original Tudor house that had been dismantled in the late 1930s before being 'rebuilt' by a master builder in a different location with Arts & Crafts additions. When we purchased it in 2007 the vendors had no idea of its origins and it was being marketed as a 1930s 'Tudor style' property. It was only when we delved into the history of the area we discovered the truth.
This had beams everywhere, some original, but most dating from the 1930s. We kept everything in tact as it was utterly beautiful and hard to distinguish where the old stopped and the new began.
The second house was originally built as a three-storey detached cottage in the early 1850s. In 1924 the then owners added a fantastic two-storey extension in Arts & Crafts style with ornate brick/stone fireplaces and exterior detailing such as a characterful verandah, at the same time applying beams to the victorian ground floor ceilings and herringbone parquet to the floors as well as Crittall windows/external doors throughout.
We did paint the ceiling beams in two rooms of the second house (same colour as the ceilings - Craig & Rose Pale Oak), but left them as they were in the large main living room. We wouldn't have dreamed of removing them as they formed part of the house's history, plus we love the Arts & Crafts architectural period.
@cocobelon It looks like a lovely property and I think with some research into the Arts & Crafts Movement style (and Charles Rennie Mackintosh's Art Nouveau) as suggested by previous posters, you can enhance and improve what you have.