Hi
If you are planning on painting it, ultimately, it is best to remove the remaining residue.
You could paint over it but as you say, depending on what paint you use, as you say, it may react (hard to know without knowing exactly what paint and what the adhesive is). Additionally, by painting onto the remaining adhesive you will be bonding the paint to this and if it becomes loose the paint will come off also.
In terms of removing, there are several options - You can but specialist adhesive removers (should be available from your local DIY store), you can scrape it off using a scraper or hammer and bolster if it’s tough, soften it using a heat gun or wallpaper steamer etc….. Some tips on this can be found in the following project:
https://www.diydoctor.org.uk/projects/Removingcork.htmOne thing to be aware of is that heating adhesive can give off harmful fumes so be aware of this and make sure you wear a suitable mask to prevent breathing the fumes in and also keep windows and doors open to allow venting.
Hope this helps