This is a very common question that we are asked when a tooth needs to be taken out. It is definitely possible to leave the space with no replacement. For some people there will be no major issues aside from a decrease in chewing ability or aesthetic concerns, but there are some problems that can also arise over time.
The main issue at play is the natural tendency of teeth to drift towards an open space, particularly forward and outwards. Over time this can lead to a situation in which the teeth no longer line up properly between the upper and lower jaw. Problems with chewing can follow and in some cases even TMJ (jaw joint) issues can arise.
Also, surrounding and opposing teeth that have tipped and moved can be much harder to keep clean. Tipping quite often leads to areas of the roots which become exposed. These teeth are at higher risk for cavities and gum disease over time.
When a tooth has been removed, there is a window of time that is ideal if a replacement is planned. Waiting too long can make treatment much more complicated. Options such as dental implants, fixed bridges, or removable tooth replacements are all possibilities for replacing missing teeth.
Your dentist is the best person to discuss this with and will be able to advise you as to the pros and cons of different options for your particular circumstance.