Most would put Tudor England firmly in the Early Modern period, so your question is slightly off. But it does present a very interesting opportunity to look at the transition as England moved from medieval castles, which were relatively open, to palaces and royal courts.

You mentioned castles housing 'the court', presumably you mean the royal court. Anyway, for most of the Tudor Period the royal family was based largely at the Palace of Placentia in Greenwich, which was built by Henry VI regent (Humphrey, Duke of Gloucester - 'the son, brother, and uncle of Kings'), and three Tudor monarchs were born there and it remained the Tudor heartland - but was demolished following the Civil War and The Restoration (today only the Stuart Queen's House remains.) Placentia was extremely private - built for exclusivity and discretion, with the option of hosting grand fetes and tourneys for the great and the good. And we see this increase as time went on: Placentia was separated from the Tudor hunting grounds of Greenwich Park by a public road, when the Queen's House was built it was designed to straddle the road - allow the royal family and their guests access to the park without crossing the street. It was this Stuart addition that finally did away with any public access to normal life at Placentia.

You mentioned castles playing host to court, which is certainly something the Tudor monarchy took advantage of (the Stuarts, not so much - the Civil War did a lot to damage the royal trust.) Elizabeth I especially recognised that she could pass much of the cost of courtly upkeep on to a hapless noble by visiting his castle/palace and not leaving until all the good food had run out - an extremely costly experience for the hosts, who would also be expected to bring their home in line with royal expectations (but the potential to curry royal favour and make deals, as well as allowing access to the court to your own allies in exchange for favours could be invaluable, if properly exploited. Either way, it strained Elizabeth's relationships with many of her nobles.) And yes, she would expect them to make the best rooms and furniture available to hers and her court, which actually is a very medieval tradition going bacj centuries.

So really, the Tudors represent the last stage in the transition away from 'castles' in the traditional sense as military and community centres. Starting with Placentia as a walled Palace with a walled hunting park, and finish with the Queen's House; which, not coincidentally, was the first building in the classical style to be constructed in Britain.