The effect of continuous rain on plants depends on what type of plant it is. There are some plants that can cope with high levels of rainfall such as those that are found in more tropical countries. Whereas those that are used to dry conditions are likely to struggle and die. The effect of rainfall on plants depends on many factors. Some of these factors include foliage cover, soil type and soil permeability (the ability of a soil to hold water), climate, leaf size, type of rainfall and many more things. The most important factor is soil as this is needed for plant survival.
The soil needs plenty of water but excessive rain can leave it waterlogged which will reduce the nutrients that the plant needs. This will lead to the plant struggling to survive or dying. However, the soil can adapt reasonably quickly and plants, especially those that flourish in the rain are likely to survive.
The amount of foliage cover will affect the plant involved. For instance, if the plant is covered by tree canopy then the effect of the rain on the soil will be less. The trees will hold the rainfall and when the rainfall capacity is exceeded on the canopy then the water will be released onto the soil. This will help reduce damage to the plant as the soil will have had time to rid its extra water in time for this new downfall and so the soil will not become clogged up to quickly for the plant to cope. However, if the rainfall is heavy and continuous it will speed up this process. If the plant is situated in an exposed area then the damage of continuous rainfall is likely to be larger. However, this depends on where the plant is situated. For instance, a plant that is situated at the bottom of a hill will gain even more water than if situated at the top of the hill where excess rainfall will run off.
The climate is particularly important. For instance, in tropical areas, plants will be used to the heavy downpour but if these plants have been introduced through buying e.g. Through a store then they may not cope. If these plants are in countries that are not used to heavy rainfall then there may be more extensive damage such as leaf damage or death of the plant.
If you have planted the plants yourself and are worried about waterlogged soils then an option could be to dig up the plant and place it somewhere where it will not become waterlogged.
Overall, the effect of continuous rain on plants really depends on what type of plant it is as this will determine how much rainfall may impact it.