Delaying field research quests

Yes, it is possible to not turn in completed field research and obtain the stamp on different day.

I tested this myself just a few days ago, knowing I would have not had the chance to complete a field request the next day. I had already received a stamp for that day, so I completed a field research but did not turn it in. The following day, I was able to redeem my reward and received the stamp.

This is also confirmed on PokemonGoHub

You can delay picking up rewards to get a future day stamp after midnight

You can keep the Field research when completed turning them in when needed for stamps. Completed research fills the page and you can’t get more tasks until you’ve turned in the fulfilled ones. The special research self completes when fulfilled.

In addition to influencing on which day you get your reward stamp there is a second mechanism that you can influence:

Certain pokémon rewards can have a special form for certain timeframes. For example, sometime early this year Eevee and Pikachu were available with a flower chain on their head during a special event. There were research rewards that awarded Eevee or Pikachu during that event and they would give the flower chain version. However, I saved one Eevee to collect after midnight – intending to get a further day’s stamp but which also coincided with the end of the event – and found it not to be wearing the flower chain to my severe disappointment.

While the story above shows why one might not want to keep quests lying around, there is a flipside: Spinda is available as a different form regularly and is only available via field research. However, its field research is rather rare. By some lucky chance, I got two of them last week and now I am keeping the second research until next month when I will be able to get a new, different Spinda pattern.