I would like to know if there will be restriction in playing most of the Beatles and Carpenters songs on a 61 key keyboard due to limitation of the keys.
Yes, you'll be able to set a piano sound and play the printed song copies with little, if any, modification. Or you could use more of the capabilities of a keyboard - maybe split the lower half to play a bass sound - and interpret the music more freely.
Even though Karen Carpenter (still my most favourite) had a range of over four octaves, she didn't use it all in any song! Most songs that can be sung will have a range of a twelfth or so (octave and a half), leaving plenty of notes above and below to play the accompaniment.
So, yes, it's easily possible to do. Eventually you may feel the need to octave notes in the bass and top parts, but until you actually start doing that, 61 keys will suffice. It's five octaves in total, so even given the restriction of root notes/ highest tonic notes in some keys, it's plenty to go on. And, you'll find on a lot of keyboards with 61 notes, there is an octave button to go up or down 2 or 3 octaves, and coupled with a key change button, the range is as good as an 88 noter.
I use a 61 key keyboard and am usually annoyed that it doesn't have ~5 more keys. That said, it's totally possible. You can play the song with probably even a 41 key keyboard, that doesn't mean you'll get the perfect balance of octaves, or any bass and treble mix at all. Replicating all of the songs by them perfectly on a 61 key keyboard probably can't be done unless you have really cool foot pedals, or a third hand and super high intellect.