1. Are both the washer and fridge 120 volt ?
2. Can you ascertain what the amp draw of each - under full load...not "running amps"
3. Once you know the combined number of full load amps - you may have options.
4. You may be able to upgrade the breakers on that circuit to accommodate the amp load...*IF*
5. The wiring that runs from the breaker box to the outlets is of a gage that is sufficient for said up grading of the breaker, and still allow 20% headroom.
Daily unplugging/re-plugging your fridge may shorten the life of the compressor slightly. Compressors consume the most electricity - and are under the most strain - when they start up. I am sure you have heard the additional sound coming from a compressor on start up.
I do not know of any risk to your house....as long as the breaker senses overload - it will pop and protect the circuit. HOWEVER...you really should have an electrician come look at it. It may be as simple as running another circuit to accommodate the fridge - given that you have room in your breaker panel for another circuit. It is worth doing and doing the right way.