If your His-tag is at the Cterminus, just insert a stop codon before it and now you have tagless protein in that same vector. If it's at the N-terminus, there may be a restriction site upstream of the His6 sequence that you can cut and re-ligate your insert to remove the tag.
If you can use tags other than His6, that would certainly help out the purification process. Other popular ones include strep and GST tags. Strep tags you can make by just encoding it into a long primer in frame with your protein.
If you are looking to express your protein without any tags, yes you can do that and there are all kinds of purification methods you can try. It will be a lot of trial and error to find the right method tho. Typically you start with isolating the cell free extract (assuming it is a soluble protein) and loading that onto an ion exchange column (check whether your protein is positive or negative charge at your buffer pH to know whether to use anion or cation exchange.) Elute with a gradient. Subsequent columns can be a combination of hydrophobic interaction resin and gel-filtration. Sometimes ammonium sulfate fractionation can help. Hydroxyapetite chromatography could work as well (though not used very often). Affinity chromotography is also a great strategy if there is a known ligand or interacting protein, though you are likely to have to make your own resin from activated resin. Not hard, but an extra step.
I assume you've done this, but at the risk of stating the obvious the easiest solution of course is to check literature on your protein to see if someone else has already worked out a purification protocol.
Also, if you are avoiding the His6 tag because it could interfere with downstream applications, you can always express it with a cleavable His6 tag then remove it with TEV, Prescission Protease, etc.