Depends on where/what your POD is...
If your not allowing a major event as POD, such as a SATURN blowing up or losing a crew or a major hardware failure on Apollo 9 or 10... it's hard to keep NASA from not making the 31 DEC 69 deadline. Assuming an abort on Apollo 11 (they came very close OTL), NASA could, and had plans to push the launch schedule left. AS-11 aborts JUL, AS 12 (Conrad) would go in SEP (maybe even AUG), they they fail, AS 13 can go as early as NOV (interestly if NASA is still reaching for the first landing with AS 13 or after I suspect you'll see a hand picked crew with McDivitt or Borman commanding.) I've even seen stuff indicating they could stack and launch AS 14 in late DEC if they really had too. BIG NOTE, this assumes the failures are all because of the learning curve not hardware or a major issue with the systems. Also most of the Manned Flight guys at NASA thought it would take two or even three tries to make the first landing.
BUT to the question; what effect.
1. If it is a series of crew aborts, for various learning curve reasons, with no deaths, no "crisis" events, NASA evenfully gets there, the program still loses support once the objective is reached. It if takes till 14 or 15 and mid-1970 the program is allowed to complete (flying AS 18 and 19) the whole program. The Soviets make big hay, but since it is very doubtfull (with their own major POD) that they would even get into lunar orbit, their propaganda only plays with those who already oppose either America or the Space program.
And...
Maybe with a few bumps in the road the US local become for comfortable with the fact that it is a hard thing to do, and cuts NASA some stack later, instead of expecting prefection every time.
2. If it is a major failure, crew (or crews lost) while I still think the program would have been allowed to try to succeed, after that game over. NASA Manned program would be seroiusly curtailed, maybe even unmanned. For the Soviets a major victory, hammering US/capitalist tech, wasting lives, money... bad all the way around.