Commonly more than you want to pay. The structural bending strength of a 16' door beam has to be 4 times that of an 8 footer, meaning typically double the beam depth in normal construction. That is because the moment (bending force) is proportional to the floor load above times the length squared, not just proportional to the span.
Unless your garage was built with single and double span as options, then a big enough wood beam to span both lanes would probably be deeper than your available space, so a steel beam, multiple parallel beams with new posts intruding into the garage space, or a flitched (composite steel plate with bolted wood beams on each side) beam might be required. Some garages are built with this center post option as an architectural detail and actually use a 16' span beam for both, so in that case it would just involve tearing out the center post and a bit of siding and paint touchup, for about $500 or so.
If not built with a 16' rated beam, then a portion of siding and ceiling drywall would have to be removed, the entire side of the house over the garage doors would have to be supported on a temporary wall, the garage doors removed, the beams and center post removed, the outer posts beefed up, the new beam installed, floor joists tied into it, door jamb and trim pices installed, siding and drywall replaced and painted, and a new garage door installed.
On a recent job just replacing an existing 16' span broken beam (no Virginia, an F350 pickup with offroad tires and lift kit will not fit into a normal size residential garage, even if you hit it fast) cost a bit over $3000. To replace two doors with one and a 16' span beam would probably be about $5-8,000, I would guess, depending on whether you have room for a standard wood beam or need to go to steel.
If you are still considering going ahead with it, you need to have a structural engineer dealing in residential construction evaluate the loads, then design and provide stamped drawings for the modification, as this will be needed for the building permit. Probably about $350-600 initially for a site visit and measurements with evaluation and determination of whether the existing beam can handle the full span, and if not then about $300-600 more for a design to fit your conditions. Note he will almost certainly have to cut a couple of head-sized drywall holes in your ceiling to see what is there and get measurements, so that repair cost if the job is not gone ahead with would be about $200 or so. No noticeable added cost if job is done, as hole repair can be done along with the replacement job.