I was wondering if it's OK to use the data presented in the form of Tables in other publications and cite them as the source of data? Like doing new analyses on these data? Should we ask for permission from the authors or just citing them would be OK?
It is absolutely okay to use data from another publication for your own analyses. Indeed, this is one of the main reasons why often raw data must be made public (unless there are specific contractual or ethical reasons not to). Once the data is published it is in the public domain that there is no requirement to ask permission and the original authors have no right to object to you using it - if they wanted other not to use it, they shouldn't have published it. You do need to cite them however.