This all depends on the weed you leave in your garden. In general the main thing that can happen when weeds are left in the garden is they will compete with other plants for nutrients and space. This is why it is important to keep a close eye on the weeds you do choose to leave in your garden.
However, there are some plants including japanese knotweed that should always be removed from a garden as it is a non-native invasive speices. It is also an offence to cause it to grow in the wild under the provisions made within Schedule 9 of the Wildlife and Countryside Act 1981.