It's an easy mod for a woven tank pattern:
1) When cutting out the front and back of your tank, also cut out a front and back in a featherweight cotton muslin (or you could use a polyester lining material if you want it to feel cool and slippery).
2) Trim about .75 inch to 1 inch off of the hem of the lining front and back. You want your finished lining to end up a bit shorter than your tank. (If your tank pattern calls for french seams, trim the extra seam allowance off of your tank and lining pieces and just go with regular overlocked seams since your lining will hide your seams on the inside)
3) Proceed to follow the pattern instructions to sew your tank together. Whatever you do to sew the tank, do the same to the lining, including finishing the seams. Then hem the lining in just the same way you hem your tank. Now you have two identical tank WIPs, one in your real fabric and one in the lining fabric that is slightly shorter at the hem.
4) Turn your tank wrong side out and your lining right side out. Slip the tank inside the lining so that the wrong side of the tank is against the wrong side of the lining.
5) Pin them together at the neck and armholes, lining up the shoulder and side seams. The raw edges should be aligned. If the lining overlaps in some spots give it a little trim.
6) Machine baste them together at the neck and armholes within the seam allowance.
7) Now you can turn the whole thing right side out and continue on with the pattern instructions to add the bias binding. This photo shows a finished armhole:
The finished hems will be offset by just enough to keep your lining from peeking out:
Now you can wear your lined tank with confidence, and keep your underwear color to yourself.