Is it possible to pay off the balance of one credit card using another card for the purpose of moving debt between cards indefinitely? Assuming this and that no changes to the amount owed are made(no further debt added or paid off), are there any consequences associated with this activity?
It is impossible. Even with 0% APR and no balance transfer fees, you're required to make minimum payments. Thus, you would have to reduce the debt. The minimum payments are often a percentage of your debt (above certain limit), so if you want to roll the same debt around, you'd have to spend additional money, because you'd be forced to pay some of it off.
Perhaps. Each card will (a) be a credit inquiry which will quickly impact your score. The inquiry stays on your report for 2 years, so if the new card had a 1 year zero rate, your report might just show 2 credit inquiries. The new cards would drop your average card age, which is also bad. It's a question of whether you keep the card open once you are done with the deal or cancel it.
For the record, the above are my personal two weak areas. Refinancing 2 mortgages, and accepting two cards with terms too good to decline cost me 30 points on my score, but the savings were worth it, and the score is slowly returning.
The biggest risk is that the current deal might be coming to an end and you have no new card ready. The rate goes 'normal' and you're at 18%. These deals typically have a transfer fee of 2-3% with no limit which is still an expense.
So, yes, possible, but it's a dangerous game if you have no "plan B."