Rev Michael Long from the Notting Hill Methodist Church said the amount of people that had turned up to drop items off had been "powerfully overwhelming" and urged people to help in different ways.
He said: "Some people have carried things a long way - they've brought things in cars and lorries and it's very frustrating for them if then they find nowhere that they can actually then deliver it to."
He instead urged those wanting to donate to "simply hold people in their prayers".
Community centres and churches became overwhelmed with donations with some centres being forced to turn people's offers away because they run out of space.
Speaking about the importance of prayer during this difficult time, Rev Long said that while it may be hard knowing what words to say to God, people should pray that "some sense of peace for those that have suffered so much might emerge".
Dozens of people gathered for a vigil outside his Methodist Church on Wednesday evening following the west London fire that left at least 17 people dead.
Rev Mike Long invited those present to light candles to place on the steps of the church and said: "There are times when all the words we can say are not adequate and sometimes words fail us because no words can do justice to how we feel, or what we have seen or what has happened. Today is one of those days.
"What we can simply do is look to all that we have seen today which is good, which is fabulous - people getting together."
He added: "Let light triumph over all that is rotten, that is desperate and that defies our understanding."
JustGiving has said that £1.65m has been raised so far through several donation pages published on its website.
Listen to Rev Michael Long from the Notting Hill Methodist Church speaking with Premier's Alex Williams:
