Radiative Heat Balance
The earth absorbs 0.6 W/m2 more than it emits - out of an average global flux of around 250 W/m2. The total excess input power is 0.6 W/m2 * ( 4 π 63710002 ) m2 or 3.06e14 W or 306 Terawatts. Times 31557600 s/y that is 1e22 Joules per year.
How much of this is stored as biomass? Most biomass is recycled, consumed, and turned back to heat. We are burning coal and cutting down trees far faster than nature is making them, so biomass is diminishing.
One human activity that preserves biomass is wood construction. The world harvested 425 million cubic meters of wood in 2006, the latest numbers I have. Let's pretend that is all energy-dense hardwood, 1500 kg/m3, and that all of that has an energy density of 16 MJ/kg . The energy stored in construction wood cannot exceed 425e6 * 1500 *16e6 or 8.2e18 J. 1e19 joules, about 0.1% of the incoming energy difference - about 600 microwatts per square meter. In actual fact, some wood is burned as fuel, some wood is scrap, some wood buildings burn or decay, so the amount stored annually as wood in construction (or papers in file cabinets) is much less than 0.1% of the energy difference.
This does NOT include the heat we make industrially or agriculturally, but those are in the tens of terawatts. This is relatively trivial.
If all the excess input power was deposited in the atmosphere, and not deposited in ocean or land or ice and not radiated back out, it would slowly heat it. The heat capacity of a kilogram of air is about 1 kilojoule, and the atmosphere weighs 5e18 kilograms, so the heat energy necessary to raise the atmosphere's temperature 1 Kelvin or Celsius is 5e21 kilojoules, and 0.6 W/m2 or 1e22J/y would heat the atmosphere by 2 Celsius per year.
The atmosphere does not heat that fast, because it deposits energy in land and ocean and ice which are being heated, and they have a lot of heat capacity. But not infinitely so - if they get warmer, they will stay warmer, and become less effective as a heat sink over time.
When someone talks about the incoming energy difference ending up as permanent biomass somewhere, assume they are innumerate. The sums must balance over time.
Note for the innumerate opponents of those innumerates: Some worry about the ice melting in Antarctica. That ice accumulates because Antarctica is polar (the sun is at a low angle) and high altitude. If the atmosphere becomes hotter, it will absorb more moisture, and dump it in the middle of Antarctica - one of the fortunate feedback mechanisms protecting the world's oceans. Heating the oceans will make them expand, but heating the ice floating in them cannot raise sea level at all - an ice cube melting in a glass will not raise water level (try it).