This is a honed travertine. I don't think there is a way to make it shiny.

I disagree that you cannot make it shiny.
I am pretty sure there are floor finishers who can come in and polish it, similar to how they polish concrete floors, and wood floors, only using different sanding discs and polishing compounds.
In the New Canaan bath renovation project I complained to the contractor how the floor tiles were set with too much lippage (the vertical dimension of transition from one tile to the next), but that the polished tiles looked great. He took it upon himself to bring in a floor sander who ground it all even. To my horror, and that of the client's, he also eliminated all of the shine and polished surfaces that reflected the light through the windows so wonderfully. After a month he came back with a tiling expert who showed how to get the polished appearance back using a floor power sander with large discs (24" diameter or so) over the entire surface using higher and higher grit polishing compounds and pads until we reached what ultimately was as good as they could get it. Which, was close but by no means perfect. A small amount of lippage is preferred, as it reflects the daylight at different angles so that the flooring sparkles. The sparkles were lost forever, but we were at least lucky enough to get the shiny appearance back.
Lippage is defined in the American Tile Institute manuals. It should not be confused with tile warpage.

Good luck!
Dvisionone

For those of you who actually do flooring, how do you polish travertine floors (once installed) and get all the way out to the edge - like at brick or rock fireplaces, at cabinetry, door facings, etc.? When wood flooring is sanded down and re-stained we generally see about 1/8" lines around the edges of some things where they can't get it sanded down fully (even with hand sanding). Wouldn't the same things happen if you are trying to polish a honed floor? How is it dealt with?

Im wanting a tile that shines. I am wanting to install honed travertine. Sometimes it looks shiny sometimes not. Once its sealed will it shine?

Honed is a term for stone that is like chalkboard. Smooth but unpolished. What you want is to ask for polished, where they will do it at the factory.

A side note: you can field polish stone and stone tiles, but depending on what size tiles, you may not get what you want. I did a master bath with custom basket-weave contrasting marbles out of 1x2 tiles that they delivered in 12x12 mosaic sheets. When we laid them down it looked fabulous. When they finished installing they again looked great, with all of the slight imperfections in the evenness of the setting bed of thin set, when the daylight came in through the windows they sparkled like polished diamonds. We complained to the contractor though, that the variation in what is called "lippage" (a term for how much difference dimensionally exists between tiles on a flat surface. if you took a straightedge and laid it across a foot or two feet of surface, there should be minimal "lippage". The Tile Council of America has standards for what is acceptable.) We complained that it was too excessive, some were so different from adjoining tile edges that you could feel the sharp edge of one tile to the next. Before we knew what he was going to do, the contractor had hire someone in to come and sand the entire floor like you would do for wood floors. Unintentionally but stupidly, the contractor had changed the tiles from polished to honed. Putting a polish sealer on it made no difference. We ultimately had to hire a stone finisher to come in with different polishing grits and keep on going over the entire room until it looked glossy again. The problem now is that we have lost forever the acceptable level of lippage that give it the light refractions that made it sparkle. And, it was so slick when wet, it was dangerous.

Make sure your contractor knows how to communicate, and make sure you are clear in telling him, asking him if he has done it before, see samples, etc.,
The only real solution should have been for him to remove the dozen of so sheets of mosaic that were thin set and had excessive lippage, clean the bed, and reset them with new sheets. We had told him to do this and he thought he was clever and quicker doing what he did.

Never sand a tile floor!