You can line with a heavy duty plastic fabric to prolong the life of the wood. Or paint with an eco-friendly wood preservative.

I have seen plastic liner boxes available for some of the cedar planters sold at the big box stores. You will need to drill some holes in them. I make my own (small) cedar planters with no lining. It lasts for 3-5 years outdoors. The bottom gives up the fastest and I just replace the bottom unless the sides are also rotted away too much.

I am creating my first cedar planter boxes and have decided to line them all with landscape fabric. I'm hoping it will keep the moisture off the wood and prolong their life somewhat. I have left a small gap (1/4" or so) between the boards in the bottom and am hoping the water will drain between the landscape fabric and the wood and out the slots in the bottom.

Landscape fabric will have little to no effect in keeping moisture away from the wood. It is intended to be fully permeable, otherwise it would prevent moisture from reaching the roots of plants when used in a garden setting.

Will those heavy duty plastic drop cloth be okay?

Get the 4mil or even better 6mil plastic sheet. I think what they call heavy duty drop cloth is about 3mil. The thicker ones are easier to staple and last longer. You would still need holes in the bottom for drainage. Ideally, you should cover the top edge of the box too. Otherwise, water can easily seep in between plastic and the wood and stay wet there for long duration.

Will something like this work?

https://www.amazon.ca/gp/product/B0026T3Q2W/ref=s9u_simh_gw_i1?ie=UTF8&pd_rd_i=B0026T3Q2W&pd_rd_r=076aa906-3a82-11e8-ba50-2f42e5cb9555&pd_rd_w=wEI6M&pd_rd_wg=Ailvw&pf_rd_m=A3DWYIK6Y9EEQB&pf_rd_s=&pf_rd_r=P37616XKAC8N4AFACBWK&pf_rd_t=36701&pf_rd_p=07871b8e-d32a-4963-b98f-ba712077f7f5&pf_rd_i=desktop

Yes, but make sure you have adequate drainage holes. That will allow some moisture to soak into the wood but but not as much as if left in direct contact with wet soil ;-)

I visited the link and it said 6mm thick and I was astonished. Anyway the amazon US website correctly lists it as 6mil (0.15mm) thick.