Removing DNA from a cell

That depends on the status and activity of the cell. Red blood cells purge their nuclei and DNA contents and continue to perform their function quite well, though they lose their ability to split and form new cells (i.e. are terminally differentiated). Most cells, however, require the genome to transcribe and translate genetic information into protein to maintain their metabolic and functional activities. As you say, DNA would also be required to divide and build new cells, so that would be one major downside of removing all DNA. Adding exogenous proteins or amino acids will not help with ensuring correct protein production and maintenance, because proteins are coded for by mRNA which act as mobile transcripts of DNA. They are dependent on DNA transcription, without which they cannot exist. Proteins do not spontaneously form. Long story short, DNA is almost always absolutely essential, and it is the case that if you removed the nucleus from most cells, they would die very shortly thereafter.