It depends on the type of test.
If they are verbal and non-verbal reasoning, it does help to be familiar with the type of test, as they will be very different from what the child has seen before.
If the tests are, say maths and English, I would suggest that a tutor who specialises in the specific school could help, as they will know the type of questions that come up. These are likely to be very different from the work done at school. (I have heard of simultaneous equations being included, for example)
And it depends on what you mean by tutoring.
If you mean going to an outside tutor, I would say it may not be necessary, but if you mean doing some familiarisation work at home with practise tests, then, as others will have done at least this, it probably is necessary.
Some people I know had their child go to short courses to learn about exam technique. These children are unlikely to have sat any kind of public exam beforehand, so this could be useful.
I think it's unlikely that an "over tutored" child would get into a school, and then struggle. If they are bright enough to respond to the tutoring, they will be bright enough to keep up with the work at school.
The only children I have heard of struggling at DD1's school are girls who are dyslexic, because the school does not cater well for SEN. Also, a lot of children come from being the brightest in their junior school class, to being only average in a grammar. This can knock confidence.
So please do consider whether it's the right school for your DC, before you do any tutoring. They may be happier at a different school, and you can use at time and money for something else like music or horse riding or something else that will enrich their lives.