Such regions are called low complexity regions and are masked by tools such as DustMasker (or dust in the old blast) in the NCBI Blast+ toolbox. Restriction afaik: works only on FASTA not FASTQ which is the only restriction for use with Illumina reads. Unfortunately, I don't know of such tool working on FASTQ data.

Edit: Well not knowing bothered me too much, so I searched for the answer. And again it's in Bioconductor, where else?

The ShortRead package contains a method to create user-defined short read filters using srFilter. One of these filters is dustyFilter(threshold=Inf, batchSize=NA, .name="DustyFilter"). That works on every ShortReads object such that you can import/export your files using readFastq and writeFastq.

prinseq-lite, which is a fairly efficient command-line Perl script, supports two methods (dust and entropy, with adjustable thresholds) for removing low-complexity reads from your FASTQ files. It also has a separate option to trim poly-A/T tails, without discarding the entire read.