Is it true that if Jews get tattoos they can't be buried in a Jewish cemetery?

No. This is a myth that arose in the 20th Century.

In Reform and Reconstructionist tradition, any Jew or non-Jewish family/spouse/partner can be buried in a Jewish cemetery, regardless of tattoos.

Among Orthodox Jews, also no, even though is a prohibition against tattoos that stems from Leviticus 19:28.

"You may not make cuts in your flesh in respect for the dead, or have marks printed on your bodies: I am the L-rd." But this doesn't prevent burials.