I recently trimmed some old dead limbs from a cedar tree in my yard. In some places, I skinned the bark with the saw, and sap is beginning to seep through. Should I be concerned? Is there anything I could apply to seal the sap in until the tree can heal itself?

Yes you should be concerned, about your tree pruning. Trees do not heal, they generate and occlude. Nothing can be done now and the wounds sound insignificant (unless you have removed the branch collars). If the resin dripping down the stem is unsightly you could try washing it off with warm soapy water, but I have never known anyone to do this.