The ACA, aka ObamaCare, was specifically designed to be paid for using measures other than a straight income tax increase, which was due to one of Obama's campaign promises not to raise income taxes on the working class (sub-250k). As such, the 85% of Americans who are insured through their employers will see little, if any, change in their 1040s for TY2014.
Assuming that your employer will continue to offer healthcare benefits after 2014 (a bigger if for some), the only change in your taxes is that your W-2 will reflect the fact that you are insured, as well as showing the amount of deductible premium payments made on the plan over the year. There may be an additional form sent to you directly by your insurer that will be attached to your return along with your other proof of income/expenses. You should not see any increase in your taxes owed between 2013 and 2014 (you have likely seen an increase in withholding between 2012 and 2013, as some of the Bush tax cuts and some payroll tax subsidies were allowed to expire).
The flip side is that, since you already have employer-provided healthcare, you probably won't be able to reduce your tax bill any in 2014. As the law stands now, tax subsidies for healthcare only apply to plans purchased on the healthcare exchanges by people whose employers do not offer healthcare, or for whom the employer plan is deemed "not affordable", so you'll only be considered for tax credits if the individual rate of coverage for the employer plan is more than 9.5% of your gross income (the threshold of "affordability").
On top of that, there's an unfortunate loophole in the ACA regarding subsidies called the "family glitch"; the "affordability" of an employer health plan is based only on the cost of the employee-paid portion of the employee's own coverage. This is typically the cheapest portion of a family's health insurance plan; coverage for spouses and children can drastically inflate that cost, but will not make the employee or family eligible for subsidies on an exchange-provided plan.
It's unlikely the glitch will get fixed before the individual mandate kicks in on March 31 2014; making such a surgical change to the law will pretty much require Democratic control of both chambers and the White House, and the earliest opportunity for that is 2015. However, by executive order, Obama has instructed the IRS not to levy the tax penalty (in effect waiving the individual mandate) on uninsured spouses/children in a household if the filing shows that the employer-provided plan would cost in excess of the affordability threshold. So if your employer plan is untenable under the new rules you don't have to have insurance at all (but ironically, uninsured families of covered wage-earners were one of the demographics the ACA was supposed to bring into the fold).
Here's a short and probably incomplete list of other things that might affect what comes out of your paycheck, whether you figure it on your 1040 or not:
Over-the-counter (OTC) medicines are no longer accepted as tax-deductible medical expenses, including for FSA, HSA and similar tax-deferred spending. In 2012, they were accepted as tax-deductible when a "letter of medical necessity" for these items was provided by a doctor or specialist (you could have gone to your PCP with a shopping list of drugs for your medicine cabinet and get a LMN covering them all). This will no longer be the case starting in 2013, so you cannot use your FSA to pay for these, and you may likely reduce your FSA/HSA contributions accordingly, increasing withheld taxes.
Persons using an HSA or Archer account for ineligible medical expenses will pay a tax penalty of 10 to 20% on the amount of the expenditure, so be careful when whipping out that debit card at the pharmacy.
The threshold for medical expenses as an itemized deduction is now 10% of taxable income as of TY2013, up from 7.5%, so if you just barely spent enough on medical expenses last year and don't expect to spend any more this year, you might find your expenses no longer qualify and your taxable income takes a jump.
Brand-name prescription drugs now have an additional tax, which is directly paid by Big Pharma, but you can bet they'll bump up prices across the board to compensate, so make sure your healthcare plan for 2014 includes good prescription copays.
"Pay to Play" Tax: Insurers themselves will pay a percentage of all premium revenue, based on their market share. This will create a balancing act; obviously the insurers will want to pass this on with higher premiums or reduced services, but they have other benchmarks to meet (or face even more taxation or forced refunds) that will limit their ability to do this.
Medicare Tax Hike: Individuals earning over $200k and married couples earning over $250k will see a .9% increase in their employee-paid Medicare taxes withheld from each paycheck. If you earn less than that you will not see any change, however your employer, if they make over $250k/yr in profits, will pay a similar hike on their side as part of your payroll taxes, and that could influence hiring/firing/salary decisions (a company employing around 120 people will, in effect, have to pay an entire workers' salary to the government in new taxes for 2014).
"Cadillac Tax": If your current plan's premiums cost over $10,200/yr (including the employer-paid amount) for you individually, or $27,500 for a family, the insurer will be levied a 40% tax on the portion of the premiums in excess of this amount. The point is not only to raise money, but to encourage insurers to lower costs and therefore premiums, as these high-premium plans become a no-win for the insurer (they're required to pay 85% or better of total premiums on claims or else pay refunds, and having to pay this excise tax on a portion of premiums is no excuse for not meeting that target). The effect as seen by you is that plan prices above this threshold will balloon to the point where they're not a viable option for anyone but the richy-richest (and not worth it for them), and your company would be offered lower-cost but lower-value plans from the insurer.