Excessive pumping of groundwater can cause the water level to be lowest near the pumping site. The water flows towards the lowest point. Normally it will be a river or lake, but with heavy pumping the current can be reversed.
Yes. There is a term "cone of depression", for when a well draws down the ground water table. There is a simple graphic here: https://www.e-education.psu.edu/earth103/node/900 .
The influence of the cone of depression can include
lowering the level of a pond or river
drawing in saltwater from a saltwater body
drying out wetlands
drying up nearby (shallow) wells
Obviously there are many factors that influence the effects of pumping groundwater with a well.
Assuming you are in the same aquifer system and if you are adjacent to small streams, wetlands etc. and the pumping rate is significant enough you may notice a change in stream levels or water levels. For a big river like the Ganga or Brahmaputra it would not be noticeable or measurable.
There are multiple factors that are involved, pumping rate, frequency of pumping, distance form the surface water body, hydraulic conductivity, aquifer storage coefficient, stream bed conductance etc.
Use this page (STRMDEPL08—An Extended Version of STRMDEPL with Additional Analytical Solutions to Calculate Streamflow Depletion by Nearby Pumping Wells (usgs.gov)) as a starting point to calculate the depletion you can then get a feel if the depletion rate could cause a noticeable or measurable level change in the surface water feature ( river or stream).
Also don't forget the cumulative impact of many wells pumping at smaller rates that cause the same impact as one well pumping at a big rate.
A single well near a small stream can suck a stream dry, but several wells near a big river may not have much of an effect.
As Naba summed up, if you take out more out of the system that is coming in over time you will cause a change.
The degree of influence depends on the aquifer (aquiferous horizon) from which the groundwater will rise, as well as on the depression curve of the well. The nature of the depression curve is influenced by the geometric parameters of the well, as well as the slope of the depression curve, which depends on the characteristics of the soils by the filtration coefficient.
Within the catchment scale, surface water, and groundwater form a unique system. The water balances within the catchment area together with hydraulic bases require that all other things being equal on all the catchment, any additional abstraction from the aquifer results in a lowering of the piezometric line in the water table as well as on the surface of the river. Conversely, the same balances require that all other things being equal on all the catchment, any additional withdrawal from the river results in a lowering of the piezometric line at the surface of the river as well as in the water table.