Maybe. Depends how sensitive the boat lift is to supply voltage - #10 copper at 14 amps for 200 feet is about 5.8% drop (or 7 volts) - nearly double what is normally designed for. But given 120V at the head end, that's still 113 volts, and many items are perfectly happy with 110-125V.
http://www.electrician2.com/calculators/vd_calculator.html
Is the 3 conductor #10 OUTDOOR wire? If it's indoor Romex, it cannot be used outside, even in conduit - the insulation is not rated for water, and all conduits outdoors are assumed to be wet (and the assumption is usually right.)
If you are buying wire, look at #4 or #6 Aluminum - it may well be a LOT cheaper than #6 or #8 copper, even if you have to buy conduit one size larger.
http://www.electrician2.com/calculators/rf_calculator.html
(not affiliated in any way, just one of the more useful/less-spammy sources of VD & Fill calculators - there are many others.)
When you are trenching for the power, consider dropping another conduit (even 1/2" could work) in case you decide later that you'd like WiFi at the dock area. No need to run wire now, but you'll kick yourself later if you don't, and it won't cost much to add.
The only solution, is to install a code compliant grounding conductor, circuit breaker, and receptacle.
The circuit breaker is easy. Since you're going from 30 amperes down to 20, you can simply swap the 20 ampere breaker in. However, since you're changing it to a 20 ampere branch circuit, AFCI and/or GFCI protection may be required.
The receptacle is also easy. Just swap out the old receptacle for the new one.
Installing a code compliant grounding conductor may require pulling a new cable. Depending on the home, you might be able to find another way. Though without being on site, it's difficult to suggest a solution.
Once you have the proper circuit protection, wiring, and receptacle. Then you can buy an extension cord, and safely extend the reach of the cord.
If you end up having to pull new cable anyway, you might consider simply installing a new receptacle closer to the tanning bed.
Best Answer
I can't speak to the code issues and/or real-world safety of running a 220V 15A device on a 30A circuit.
However, keep in mind that dryer (and range) receptacles are NOT designed for frequent usage. They are designed to sit untouched for years at a time. Where a regular plug & receptacle might get used a few times a day (and with cell phone chargers and other devices sometimes more often than that), the dryer cord/plug and receptacle may not do well being plugged/unplugged on a regular basis.
In addition, you need to make sure the dryer receptacle is a newer 4-wire receptacle - hot/hot/neutral/ground NEMA 14-30. Older dryers (or rather, older houses never upgraded) use a NEMA 10-30. It is code-compliant to continue using a NEMA 10-30 for a dryer. But you should definitely not use it for a new device. So if you have a NEMA 10-30 then, in addition to any other issues of 15A vs. 30A, etc., you should be upgrading first to a NEMA 14-30 receptacle and change the dryer cord to match and remove the neutral-ground jumper inside the dryer.
Also see: What's a simple solution to have 2 NEMA 14-30 receptacles and switch power between them? for a discussion of using a switch between 2 receptacles.