There are two things happening when you input your healthcare information.
First, as long as you answer the TurboTax question saying you
had coverage the entire year, you won't be penalized for not having insurance. If you were without insurance for more than two months during the year, then you would be subject to a penalty. That penalty is one possibility for your taxes rising.
The second thing is a bit more complicated, and it involves the 1095-A figures. When you have
a policy from the Marketplace (Obamacare), you have to estimate your income when you sign
up. They use that estimate to calculate how much (if any) subsidy you'll get
to help pay your premiums (it's called the Advance Premium Tax Credit).
Then, after the end of the year when you prepare your tax return and
enter the 1095-A, they compare your "actual" income with the estimate
to determine how much of the subsidy you were actually entitled to. This
is where the final accounting of those subsidies takes place.
If your estimate was too low and you made more income than
expected, then you'll have to pay back some of that subsidy. If your estimate
was too high and you wound up earning less income than expected, then you'll
receive additional credit on the tax return.
In your case, since your taxes went up after inputing the 1095-A & healthcare info, you're either paying a penalty for not having insurance, or you're paying back some of that subsidy shown on the 1095-A. Or it could be both.
When you print out your return, take a look at the form 8962, line 29, which shows the amount of any payback you must make for the subsidy received.
