First we must determine what you mean by a dairy food processor. There are a few ways in which a food processor can become dairy. One is if hot dairy food was put in it. Another is if a dairy residue was washed from it using hot water without a significant concentration of soap. A third way is if a dairy liquid remained in it for twenty-four hours.[1]
If your food processor is indeed dairy, then you have a problem because an onion is a classic example of a davar charif (a sharp food). The status of davar charif makes the interaction between different foods more halachically significant because it negates the following three reasons for leniency (some totally and some partially) that might otherwise apply:
1. Usually taste is transferred (between two or more foods and between foods and utensils) and absorbed only when there is heat present. However, if a knife cuts even a cold davar charif, there is a presumed transfer. Thus, it must be assumed that the onion will absorb a dairy taste[2] from the processor’s blades.[3]
2. Usually, when kosher taste (e.g., milk or meat) that is absorbed in a utensil comes out into a food, its status is significantly reduced[4] because it is “noten ta’am bar noten ta’am” (doubly removed taste). However, a davar charif causes even such an otherwise weak taste to maintain enough strength that the leniency associated with the doubly removed taste does not apply.[5]
3. Usually, taste that had been absorbed into the walls of a utensil, if left for 24 consecutive hours, deteriorates and somewhat worsens the flavor of foods into which it is expelled. In most cases, if foods were cooked in such a utensil, the expelled taste does not change the status of the foods b’dieved (after the fact). If, however, the absorbing food is a davar charif, most poskim rule that it is assumed to receive a positive infusion of taste from that which was absorbed in the utensil, and may take on its status.[6]
For a combination of these reasons, onions cut in a halachically dairy food processor should be considered dairy and not be put into fleishig foods. [Permutations of circumstances can create various rulings in similar cases. The above is but a general background.]