The jsonlite package on CRAN supports streaming in of JSON data (if your file is in the appropriate .ndjson format).
See the help for ?stream_in in the jsonlite package.
Because parsing huge JSON strings is difficult and inefficient, JSON streaming is done using lines of minified JSON records, a.k.a. ndjson. This is pretty standard: JSON databases such as dat or MongoDB use the same format to import/export datasets. Note that this means that the total stream combined is not valid JSON itself; only the individual lines are. Also note that because line-breaks are used as separators, prettified JSON is not permitted: the JSON lines must be minified. In this respect, the format is a bit different from fromJSON and toJSON where all lines are part of a single JSON structure with optional line breaks.
You can also evaluate the ndsjon package on CRAN:
Streaming 'JSON' ('ndjson') has one 'JSON' record per-line and many modern 'ndjson' files contain large numbers of records. These constructs may not be columnar in nature, but it is often useful to read in these files and "flatten" the structure out to enable working with the data in an R 'data.frame'-like context. Functions are provided that make it possible to read in plain 'ndjson' files or compressed ('gz') 'ndjson' files and either validate the format of the records or create "flat" 'data.table' structures from them.