Can a cell phone be traced or tapped?

All cellphones, even old analog phone, can be tracked to some degree. Those programmed with GPS software can be tracked directly with a relative degree of accuracy.
In the infancy of the cellphone industry, it was possible to track the cell from where a phone was accessing the system (switch). From this, it was relatively easy to narrow down a rough location of a cell phone user. This can still be done today.

With modern cellphones, even if the phone is turned off, outside access is possible to determine location, provided a power source remains constant. Remove the battery, remove the access.