Although I could say hundreds of things about this old but not forgotten gem I'll skip to my question.

The game runs perfectly, no doubt about that, but is there a way to increase the gameplay resolution?
I already tried changing some settings, like "Direct3D" instead of "Surface" which was the default selection. I also set the resolution to 1440x900 to match my desktop's resolution.
But no luck, the game still runs in this old 320x240 resolution.
I mean isn't there a way to make the graphics look smoother and more detailed?

The only thing you can do is fiddle around with output and filters as the game does not natively support higher resolutions, the reason you select resolutions in dosbox is for the game to make best use of the screen.

2xsai or super2xsai are strong filters that smooth the image out a lot, disadvantage is that text can be hard to read.
Personally I use OpenGL and normal2x for most games with keep aspect ratio checked, but I don't mind a bit of blockiness.

What do mean by "blockiness"?

Jagged edges.

I know this is probably too late but if you want to increase resolution you have to edit the dosbox config file for bioforge in your bioforge directory. set fullscreenresolution=desktop and save the file and then your games should be scaled up some renders only render in a small box so pick one that looks nice and stretches the image well and put a good filter on it