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I hope this finds your dog doing well. Drano is so corrosive that animals generally do not lick up much of it. Usually the ingestion is very small and does not cause any problems. The big thing to look for is oral ulceration. Look for any red or bloody patches in the mouth, on the tongue or under the lips. If you see any in the mouth, then there are also likely some in the intestines. If the mouth is clean, then you can bet the intestines are fine as well.
It may not hurt to get your dog on some Pepcid (Famotidine) for 3-5 days. The dose is 0.25 mg/lb every 12 hours.
If there are significant oral ulcers, then a trip to the vet for some antibiotics and pain meds would be indicated.
I hope this helps, let me know if you have any other questions.