PhD programme without teaching
Definitely yes: if in the future you're looking for a teaching (or part-time teaching) position, having less experience than other candidates will be a really bad experience.

Yes to some extent: even for non-teaching research-only positions, teaching experience can be seen as a positive for the applicant. Some of the qualities required for teaching are necessary for other aspects of the job (handling a team, tutoring interns or PhD students, communication with non-scientists, etc.).

So, I would advise you to continue having some teaching duties during your PhD, even if you have to take those outside of your own institution or do some of it pro bono (or for not-so-great pay).