A basement support post is installed to keep a foundation frame in a wood-frame house from sagging under the weight of the building. Typically these posts are made of wood or steel and laid underneath beams or floor joists. They have an adjustable screw top that can be raised to fit the space between the floor and ceiling of the basement, and if the frame starts sagging, can be adjusted to fit the sagging frame better. They are designed to support the weight of the home above. They literally support tons of weight from above moving them without some preparation is usually not a good idea—unless you want the house to come down.
The post they support above can be removed only if a new and stronger beam is installed. The new beam must be designed to carry the weight of the floor above over a greater span. To determine the beam's required size and span limit, you should ask a structural engineer to calculate the load requirements. Do not do this yourself, as putting in a new beam not designed to handle the weight could cause serious damage to the home. Wood glue laminated beams are widely used for such purposes but often interfere with head room in the basement area. An alternative is a steel I-beam, which can then be boxed in to hide it. This will be a more expensive option though.