Yes you can change your STI brake levers/shifters for ones with cable routing under the bar tape. There is nothing special required.
However I see very little advantage in routing under the bar tape. It looks slightly better (subjectively) and improves aerodynamics slightly. Personally I upgraded from 105 5600 (good old “washing lines”) to 105 5700 due to a broken shifter. I have the impression that the 5700 with cable under the bar tape shifts slightly worse, which wouldn’t be surprising considering the sharp bent required.
I assume these are Ultegra 6600 STIs? They are 10 speed.
Unfortunately the only compatible 10 speed shifters with routing under the bar tape are the 105 5700. Ultegra 6700, Ultegra 8000 and 105 5800 are all 11 speed. You’d have to upgrade your rear derailleur, cassette and chain as well.
Edit: Oh, Ultegra 6700 is 10 speed and has routing under the bar tape. Thanks to JoeK for pointing that out. So you could go for either 105 5700 or Ultegra 6700.
The 10 speed Tiagra shifters are incompatible with other 10 speed rear derailleurs.
Yes, with some qualifications. I'm guessing you've got Ultegra 6500 shifters, for a 9-sprocket cassette. Current Ultegra shifters are designed for use with an 11-sprocket cassette, and Shimano differentiates its road groups, in part, by the number of sprockets they work with; the group that works with 9-sprocket cassettes currently is Sora. Here are the three options I can think of.
- If you replaced your levers with Sora levers, that should work with no further modifications.
- If you wanted to stick with Ultegra to maintain a consistent level of "fanciness" throughout your bike, things get more complicated. 11-speed cassettes require a different freehub body (the part of the hub that the cassette slides onto) with a tiny bit more width for the cassette; freehub bodies often can be replaced, but it's obviously more of a to-do. So that would require you to change cassette, freehub body, levers, chain (chain widths are different, and it's generally a good idea to replace your chain when you replace your cassette), and rear derailleur (Shimano 11-speed derailleurs have a different cable-pull ratio).
- Your current freehub body will work with a 10-speed cassette; the 10-speed generation of Ultegra levers (6700) did route the shift cables under the bar tape. So for this, you'd need new levers, cassette, and chain. You would have to buy these levers used or find someone selling them as new old stock.
Compatibility can be complicated and there are some weird edge cases; I'd read this set of articles before spending any money.
The levers in the picture are from the generation where the shifter cables were 'non-aero' (Shimano designated them that way) meaning that they ran out of the top of the brake-levers to an anchoring point on the downtube. It was only later types where the routing of the cables turned by 90° backwards to the handlebar which enabled the housing to follow the same route as the brake cable housing and to disappear under the bartape.
Unfortunately you cannot switch the routing in the way you intend. The inner workings of the levers are totally different.