The phrasal verbs dispense with and dispose of are synonymous in some of their uses, but there are other senses in which they’re separate. The phrases are also interesting in that dispense with and dispose of are quite different from their one-word equivalents, dispense and dispose.
Dispose of
Dispose of has a few meanings: (1) to attend to; (2) to part with, as by selling; (3) to get rid of, usually by throwing out; and (4) to kill or destroy. The third sense, as used in the following examples, is the most common one:
A Sheridan woman tried to dispose of evidence by eating a baggie of meth during a traffic stop north of Buffalo, Wyo., Thursday. [Billings Gazette]
The moment garbage is put out on the curb, it becomes a commodity, worth money to anyone willing to dispose of it. [Albany Times Union]
Dispose on its own means to settle or decide a matter, or to place in a particular order. Using dispose in place of dispose of to mean throw away or get rid of can confuse readers. For some reason, this questionable use frequently appears in Indian publications—for example:
Hospitals that did not dispose their biomedical waste were also penalized. [Times of India]
Dispose off in place of dispose of is another common occurrence in Indian publications. We can’t explain this.
Dispense with
Dispense on its own means to distribute in portions, to administer, or to grant an exemption. The phrasal verb dispense with is almost completely unrelated, meaning to go without, or to do away with. In this second sense, dispense with is similar to, but not exactly like, dispose of. For example, these writers use it well:
But look closer, and it’s clear that the administration’s real goal has been to dispense with Mubarak while keeping the dictator’s military subordinates very much in charge. [NY Times]
The new corporate entity will be known as NBCUniversaland will dispense with the peacock and globe motifs at a corporate level. [The Spy Report]
And these writers correctly use dispense with as a synonym of forgo:
We dispense with the qualifiers and state, as absolute fact, that Belichick is the greatest coach in NFL history. [NFL News]
Like many companies, the Government wants to dispense with owning lots of big computer systems. [New Zealand Herald]