In short no, nothing like, hunting task points, charms, bestiary, levels, skills, items, gold, quest and acces remains to the next tournament. In every tournament the character was reset.
On every new Tournament, all characters start at level 20 with basic Skills and 20,000 Gold in their bank balance.
Store products such as Furniture and Training Dummies are not lost when a Tournament Ends. The character will have these already when the join a new Tournament.