My daughter, is 3.5 in PK3 this year. Previously she went to a montesorri school, however, she started having issues once she was moved to the lower elementary age class, her teacher suggested we get her tested for learning delays/social delays as she was having tantrums in class, crying, etc.

I got her tested and at 3, she was at the age 5-7 level. My daughter also seemed bored in class and completed work quickly (which they had a problem with) as it wasn't a challenge. I moved her to a more academic program about 1 month into the school year. All seemed to be going well for quite a while, although the first week we did have a school meeting as she was again having tantrums and crying. We discussed strategies and all seemed to be well. We got interim reports recently and my daughter hit all the requested educational milestones and then some. I asked if she would continue to be challenged as she had advanced so quickly and was given a vague answer. Yesterday I recieved an email from the director of the school who recommended removing her from the extended day program as she felt she was 'too tired' because she was still having tantrums and fell asleep after them. (Instead of the tantrums making her tired?)
I realize my daughter is a challenge as she is a high spirited child, and I continue to work on her emotional outbursts on a daily basis,however, It seems as if, that if the school can't deal with her. Or any school, it seems. Perhaps it would be better to remove my daughter from traditional schooling all together, and maybe hire a nanny for this year, and/or next year and see if it is better for her.

TL;DR
Should I remove my high spirited gifted child from traditionally schooling as she continues to have emotional outbursts and hire a nanny/home teacher.

I'm not sure that your choices are binary (home teacher or keep in traditional schooling), based on what you wrote.

She is very young, despite being able to do higher level work. What she may need is less academic type work, and more "fun" but enriching work (which sometimes can be provided in a project-based, Montessori or other type of preschool). She may also need more time to play, dance, swim, do gymnastics or art, etc. Perhaps a nanny would be a good solution if said nanny is good at keeping a highly spirited child appropriately busy with a variety of activities.

It's also possible something else is leading to the tantrums, although some children just have a lot of them at that age, it can also be a sign of other issues, I believe. I'm not an expert there.

Our son struggled a lot between 2.5 - 4 yo. And when we moved him up into a class that was a better fit intellectually it was a disaster socially, so we moved him back down. I'm not sure there was a "right" fit, but we all survived and he is a well adjusted, happy 6 year old now. Every kid is different, but I think this age can be hard in general for gifted kids.

My (very personal, subjective) take away from the experience is that a play based preschool with very little "school" type work is often the best fit. Sure he was reading books at home and learning letters at school, but most of the time was actually free play, painting, etc, so it wasn't a big deal. The social and emotional development at that age was really the most important part.

I would also echo that the tantrums, etc for DS are usually a sign of something that needs addressed. If you've tried to make it work with the school for a while and you don't know of any other outside issues at home, etc. it does seem like time to reconsider the school choice.

Is the environment too stimulating? At that age my son became stressed in very noisy, very stimulating environments.

Good luck. It's so stressful, but you'll figure it out.

I would look for a preschool that is mostly free play but has an ample supply of books, a science and nature table, as big and natural or garden like playground as possible and teachers who actually talk to the kids about stuff. If I could afford it though I would go for a nanny who would spend hours outside in the fresh air, take her to activities and play games and read a lot. I would also have a reasonably predictable flow to the day and week.