Yes, it might. Some 3D printing technologies take the bounding box into account in their pricing, therefore part orientation can indeed influence the price (see also the article: How are prices of 3D prints calculated?).

When uploading your file, it is best to orient your model so that the bounding box is minimal in order to save costs.

What is the effect of the check box on OnSite: “keep print orientation”?

If this option is checked, the part will be printed in the orientation that you have uploaded the model in.

Why does a bigger bounding box lead to higher prices?

Pieces with a big Z dimension, such as a cylinder that is printed standing up, take more time to print than when lying down.

Also, some technologies require support structure which is needed to support the model during printing. The orientation of the part can influence how much support structure needs to be generated. E.g. if you position a cup upside down, then a lot of support will need to be printed inside the cup, which affects the price.

What is a bounding box?

The model inside your file holds a specific orientation in three dimensional space, which is measured in the the XYZ coordinate system. The XY plane is the ground plane and Z represents the height. The bounding box of a model is an imaginative box drawn around the outer dimensions of your model if you were to project it on the XY, XZ and YZ planes, that is: the amount of space that the part takes up compared to the XYZ axes.

For example: a cube which is aligned with the XYZ axes has a smaller bounding box than a model that is not aligned with the axes, see the example in the 2D graphic below. For a cube that would mean aligning the sides along the axes would give the minimal bounding box. Below is a 2D visualization showing that an aligned box has a smaller X and Y dimension: