I assume you are getting rid of the lawn and then putting membrane down followed by ? gravel/bark/cobbles?

Personally I'd kill the grass and weeds first then put the membrane down.

I would also 'kill' the lawn first. If you lay the membrane on top of the grass, the grass/weeds will still keep growing for a period of time and lift up the membrane.

Totally agree - you need to cut then kill the grass first. Weed suppressor are just that, they suppress plants not stop them. The grass will grow through the tiny gaps in the fabric, especially the most pernicious ones. Grass can also establish on the suppressor fabric as I have found. Bulbs can also grow through the fabric - have seen Daffodils struggle through it.