No, if you want to keep your current shifters, and go 10 speed, it's not possible; as the rear shifter doesn't have enough clicks / positions. You will probably find it easier / cheaper to get a new bike if you want 10 speed.
'''96 Litespeed Catalyst, '05 Litespeed Firenze, '06 Litespeed Tuscany, '12 Surly Pacer, All are 3x8,9 or 10. It is hilly around here!
You can keep your rear derailleur since 9-speed Dura Ace is compatible with other and newer groups. You will need new shifters as jimc101 explained and, unfortunately, brifters are the most expensive single item. You could save a fair bit of money by using 105 or Ultregra brifters unless you insist on Dura Ace.
If it's a tri-bike, it probably has bar-end shifters. Bar-end shifters usually have a friction mode, but running 10-speed friction mode takes a very steady hand to align the rear derailleur well enough on every shift to keep it from ghost-shifting. In practice, friction shifting is for 8 speeds or less. You'll need new 10-speed shifters, but the derailleurs are probably OK.