This isn't true. The actual GDPR text on consent states [1]:
the data subject has given consent to the processing of his or her personal data for one or more specific purposes;
-> So processing of data based on consent is limited to the purposes agreed to.
or
processing is necessary for the performance of a contract to which the data subject is party or in order to take steps at the request of the data subject
-> So processing of data is necessary to fulfill a request made by the data subject.
This is a far cry from "processing the data which you have signed away". The only blanket allowance for data processing is this item:
processing is necessary for the purposes of the legitimate interests pursued by the controller or by a third party, except where such interests are overridden by the interests or fundamental rights and freedoms of the data subject which require protection of personal data
Which again limits the processing allowed under GDPR: processing of data under this rule requires the legitimate business interest to be weighed against the subject's personal rights.