The "Output Short Circuit Current" of an MCP6002 operating at 5.5 V is in the region of 23 mA. Ref: Microchip datasheet. It will be less at 3.3 V.
For a TL081, Output Current (Typ) is 10 mA. Ref: TI datasheet.
So your circuit as presented is not going to approach 100 mA let alone 800 mA.
On an Arduino Due (circuit diagram), the 3V3 power output line comes through an LM2734Y with a max. output current of 1 A to get 5 V and then an NCP1117ST33T3G, also with a maximum output current of 1 A. Both devices have thermal overload and over-current protection.
It appears that you do not need to worry about drawing too much current from the power supply circuitry both because your circuit won't draw much current and because the supply is protected.
If you expect to be drawing a high current, maybe more than 300 mA, (through the power supply ICs) for a sustained time, it would be prudent to either use a separate power supply instead or add small heatsinks to IC2 and IC4 on the Due.
It is my understanding that op amps may have better performance characteristics when their voltage supplies are in the higher part of their allowed range. Adding a resistance to measure the current draw will have the effect of reducing the voltages on the power rails of the op amps, which is probably not desirable. You have 5V available so it might be better to supply the op amps with that and translate the voltages, if appropriate, for a lower overall power draw and possibly improved op amp performance.
In summary, measuring the current with a multimeter will not physically damage your circuit, but it may affect its operation and does not appear to be necessary.
If the current measurement is important then you could power your circuit from a bench PSU which has a voltage sense input. Models with a voltage sense input are going to have an output current reading. Connect the sense wire(s) appropriately and connect the PSU ground to the Due ground. If you determine that your circuit is never going to draw too much current for the Due, your job is done.