I had this same question so I did some looking around.
Here's what I picked up --
The simple formula for figuring out which tire works for you. Take the amount of contact area of the tire on the ground and multiply by PSI you have in your tire.
So, the "area" (width times length) of the tire along the ground, multiplied by the psi.
So, 2" of tire down contacting the ground (when you're sitting on it), by 1" wide tires. 2 x 1 = 2 square inches. If your tire is pumped to 100 psi, your maximum weight for that tire would be 2 x 100 = 200 lbs. That's both your bike, the rider, and anything you are carrying. (try to get an idea of whether the area down is appropriate)
Also be aware that the weight is distributed between both tires. So technically, this bike's tires should be capable of 400 lbs if you have 2 tires with the same attributes and psi.
You probably should over-specify the weight carried. I would probably spec 25% extra at least.
So, rearranging the formula with algebra.
Area x psi = weight
area = weight / psi (weight divided by psi)
So .. if you and your bike and whatever was loaded on the tire of the bike were 200 lbs...
if your psi were 50
area = 200 / 50 = 4 sq inches
You'd probably not want the length of the tire's area on the ground to be very long.. so you'd want a fairly wide tire, say 2 or 2.5 inches wide (that's pretty wide..)
4 / 2 = 2 inches for the length. (2x2 sounds light to me)
Remember that's double capacity. 400 for the whole load on the tires.
So 300 lbs of man + bike + carried supplies with a 33% over spec. (1.33 x 300 = 400)
Other considerations --
The wheels need more spokes. The info I'm giving is by inference from someone stating that a "tandem bike" (with 2 or more riders) that had a 500lb load would have "more spokes in the wheels".
Common spoke counts are around 28, 32 or 36 for "normal" applications, and for a tandem wheel it's in the 40-48 spoke range. So more spokes for more weight. There are also wheels that are "mags". Rather than spokes, they have 3 or more solid pilars from the hub to the rim. If you're really concerned about weight / strength / durability, they might be a consideration.
Then, there's the question of tire width, and whether your wheel can hold the tire width you want.
And finally, whether the wheel is an appropriate size to fit on your bike.
There are of course other considerations -- quality / metal compositions / psi ranges of tires (low and high range)..
Note that new / old / current tech can vary in durability / functionality. Just because it's new tech doesn't necessarily help in an appreciable way. (most newer tech is marginally better, except I was impressed by high end hydraulic brakes - they appear to be more than twice as powerful.) If you're not already an accomplished rider don't worry about getting high end light weight bikes. (not to mention, if you are hefting 400 lbs.. trimming 5 lbs off the bike isn't going to benefit you much..)
And so, you are further armed with information.