What should I be using to amplify a heat flux and load cell signal?
Your DAQ probably looks pretty similar to an opamp from the sensors perspective. I can't imagine your DAQ having anything overly special about its input impedance. Even if it did, you would need to be concerned about the input impedance of a new DAQ.

What I would be concerned about is the noise floor associated with your sensors and circuitry. You are needing to look at voltage ranges that most oscopes can't even see. You will need to put time into every aspect of your signal acquisition including board layout and shielding from noise.

You may be able to to use just a simple op-amp to provide your gain, but be cautious of the noise as opamps can be very noisy if you aren't careful on what you buy. There are other methods of providing a gain, but I suspect that will all have issues with noise.