It seems almost everyone is in the "no" camp, however there doesn't seem to be a lot of corroborating evidence presented.
Here's an excerpt from American Airline's Conditions of Carriage
- Carrier undertakes to use its best efforts to carry the passenger and baggage with reasonable dispatch. Times shown in timetables or elsewhere are not guaranteed and form no part of this contract. Carrier may without notice substitute alternate carriers or aircraft, and may alter or omit stopping places shown on the ticket in case of necessity. Schedules are subject to change without notice. Carrier assumes no responsibility for making connections.
Delta US CoC:
Delta is not responsible or liable for making connections
Delta International CoC
Except as stated in this rule and as provided in Rule 55, Delta will have no liability for making connections...
In the event of flight cancellation, diversion, delays of greater than 90 minutes, or delays that will cause a passenger to miss connections, Delta will [accommodate passenger...]
(Rule 55 doesn't appear to add anything substantial here)
British Airways CoC:
9b3) If we:
... cause you to miss a connecting flight on which you hold a confirmed reservation;
you can choose one of the three remedies set out [...]
2.2 Extension of Validity
If you are prevented from travelling within the period of validity of the Ticket because we:
... 4. Cause you to miss a connection
The Validity of your Ticket will be extended until our first flight on which space is available in the class of service for which the fare has been paid.
I expect most airlines will show some discretion especially if you're obviously incapable of hurrying to your connecting flight and it is at the other end of the terminal, but this discretion doesn't seem to be backed up in the CoC. There is some ambiguity in "if we cause you", since that might be possible to interpret as "we gave you such a short connection that you couldn't make it".
A few anecdotes from flyertalk. They're not directly analagous to "not running", but do show that if the airline feels it's not their fault, it might not be yours either, but you may be the one on the hook for a new ticket:
I therefore missed the connection, and they want me to pay rebooking fee and taxes and fare difference etc etc which costs like 500 usd
she was forced to buy 2 walk up Y tickets for about $1500
The agent told me that it would be more expensive to change my original ticket, than to simply purchase the additional 1-way ticket