This depends on the op-amp working with the inputs near the -ve rail. Some op-amps work well like this others do not. Check the common mode range of the inputs and input bias currents etc.
Pretty much any general purpose opamp designed for single rail use, including the common LM358, works just fine with its inputs down to and even slightly below 0V. Getting close to the positive rail is where they typically start to misbehave. (Except for rail to rail opamps, of course.)
For simplicity, you could use a NPN transistor and a pullup resistor instead of the opamp, although that would make the load degenerate to a resistor below about 0.7V rather than much lower for the opamp solution.
Then there's PTC elements, of which the ones commonly found in old CRT TVs tend to work well with a cold resistance of just a few ohms. (That's probably a bit too much of a load for your use, however.)