First of all this is a great contrasting pairing in itself. The sweet starch flavors of the fries balance the spicy aioli sauce and vice versa. Each keeping the other from becoming too pronounced.
Paired with an IPA, the starch and sweetness of the fries help to balance the bitterness of this beer. This counteracting pairing works the same way as the fries/aioli pairing; they keep each other from becoming too pronounced. The IPA bitterness actually kicks the spice up momentarily before it and the alcohol cleanse the palate.
If IPA's aren't your thing or you aren't looking to kick up the spice, then I suggest you try Hinterland's Pale Ale.