Ya what you heard is right. We should not splash water into our eyes and wash often because
- it will wash away our natural tears (these tears contain antibodies, nutrients, oxygen, mucous and a lipid layer that prevent tears from evaporating)
- it will irritate our eyes due to the chlorine content of the water
- If the water is not clean we might get infection.we might be introducing an infection if the water is not clean

 There are times when we should wash our eye with any available water:
- dust, insects, etc fallen into eye
- chemicals, paints, corrosives fallen into eye

The tears which is squeezed out of the tear glands when we blink the eye washes out the dusts and a little portion gets deposited by the nasal side/edge of the eye and the remaining got drained down to the nose through a minute channel... The eye lashes are there to prevent the dust, etc from entering into the eye...

Lime has citric acid. Any acid is corrosive to the eye. but, traditionally, this has been used in some parts of the world.

In pure dry eyes, it is infinitely better to use a drop like refresh tears, tears natural, general , etc.
General and naturallle are artificial tears drops that you can use to moisten eyes or wash dust off by applying one drop
A simple home remedy for this is keeping cotton dipped in rose water over the closed eye...