A clear sheet protector, a good seal and a laser print will keep a poster readable through bad weather. Here is how to do it cheaply.
The short answer
The cheapest way to protect a lost pet poster from rain is to slide it into a clear plastic sheet protector with the open side facing down, then tape the top and sides. Water runs off rather than getting in, and you can swap the page when details change. For a sturdier option, use clear packing tape across the front or get it laminated. Use laser prints where you can, because inkjet ink can run when wet. And check your posters after every storm. Even a protected poster fails eventually, and a blurred number is worse than no poster.
Start with the print itself
Ink is your first problem. Inkjet ink on plain paper is usually water-based, so one wet night can turn your phone number into a gray smear. Laser prints use toner and hold up much better, which is one good reason to print your outdoor posters at a copy shop and keep your home inkjet for indoor copies. If you only have an inkjet, let the prints dry fully and protect them right away. Heavier paper such as cardstock also holds its shape longer and is less likely to sag inside plastic or tear at the corners.
The sheet protector method
Buy a pack of clear letter-size sheet protectors, the kind used in three-ring binders. Slide the poster in. Turn it so the opening is at the bottom, which means rain cannot run in from the top. Fold the bottom flap under if there is room, or seal it loosely with one strip of tape, leaving a little gap for moisture to escape. Tape along the top and both sides. Choose non-glare protectors if you can find them, because shiny plastic in direct sun can hide the words from a car. Then step back and check from the angle a driver will have.
The packing tape method
If you have no sheet protectors, clear packing tape works. Lay strips across the front of the poster, overlapping each one by a little, until the whole sheet is covered. Do the same on the back, and let the tape overhang all four edges so it seals to itself. It takes about five minutes per poster, so save this for your best locations. Work on a clean table, smooth from the middle and avoid running a strip over a wet print. A home laminator gives a cleaner finish, and many copy shops can laminate a small number for you.
Mount it so wind cannot get under it
Wind can ruin a poster as quickly as rain. Tape only along the top lets the sheet flap until it rips. Fix all four sides, or use a stiff backing such as a piece of corrugated cardboard or foam board, trimmed slightly larger than the page, inside a sheet protector. On a fence, use zip ties through punched holes reinforced with tape. On glass, tape from the inside of the window if the shop lets you, which keeps the print dry and protected from wind at the same time. Always ask first on private property, and follow your local posting rules.
Choose sheltered spots
A poster under an awning, in a shop window, on a covered bus shelter board or inside a gas station door can last for weeks with no protection at all. Save your plastic-wrapped copies for the exposed intersections, which need them most. When you choose a spot, look at where the sun falls in the afternoon as well. Direct sun fades ink, especially color inkjet ink. Black and white laser copies generally hold their contrast longest, which matters more than color once a search has gone on for a week or two.
Try a yard sign for the long search
If the search continues past the first week or two, consider a corrugated plastic yard sign printed at a copy shop or sign store, with LOST DOG, a big photo and your number. It takes rain and wind far better than paper, and the sign will still look fresh a month on. Ask homeowners at key corners if you can stake one in their yard. Keep the content short, since you cannot update it as easily as paper. Do not print the last-seen date if the search may run on; use the area and phone number instead.
Check and replace on a schedule
Keep a list of your locations and do a loop after every storm and at least twice a week. Carry spare posters, tape, a marker and a damp cloth. Replace anything you cannot read from a few steps back. Fogged plastic is a sign water got in, so change the page and the sleeve. Keep a few copies of every version so you can match what is already out there. When your pet is home, go back around and take everything down, plastic included, so it does not end up as litter in the storm drain.