I read somewhere that I should not plant tomatoes close to peppers. I guess this is something to do with an increased risk of disease as they are related? How far apart do they need to be planted?

Tomatoes and peppers both belong to the Solanaceae, commonly referred to as the nightshade family. While tomato belong to genus Solanum, which includes crops such as potato and eggplant, pepper belongs to the genus Capsicum which includes both sweet and hot peppers. There are various myths and old wives tales regarding the planting of peppers and tomatoes together and perhaps you have had the misfortune to have heard one of them. The fact of the matter is that YES the plants are related and YES they share some common diseases but most people do not have the space in their garden to separate them. The reality is that because the two have similar growth requirements, they can in fact be grown quite successfully together.

Diseases common to both tomato and pepper include Verticillium wilt and bacterial spot. You can limit the likelihood of diseases emerging and spreading simply by following some good gardening practices.

Don’t overcrowd plants. It makes it easier for diseases to spread.
Don’t over or under water. A stressed plant is more susceptible to disease
Don’t water foliage. Wet foliage promotes spread of fungi and bacteria. Instead, water plants at the base.
Avoid working in the garden when plants are wet. Your movements can transfer fungi and bacteria between plants through water splash.

I would advise you to have a look at the disease pages here on PlantVillage to get yourself familiar with things to look out for. You will find a helpful table at the bottom of the plant info pages for both pepper and tomato here:

https://www.plantvillage.com/topics/t...
https://www.plantvillage.com/topics/p...

I have also included some images of the disease symptoms for some of those common to both pepper and tomato

Finally, in addition to diseases, another thing to be careful of when planting is that as tomato plants will likely grow much taller than the pepper you should be careful to position them in a manner that will not result in the peppers being shaded out.

I have always raised tomato and peppers in the same garden for years. I plant all tomato plants together and the same with peppers. I plant each in 12'x4' raised beds with the beds being 2' apart. The only problems I've encountered in many years are aphids. When they get on the tomato plants they also get on the pepper plants. As far as diseases, I have encountered none. I've always have early and late blight, to some degree, on tomato but pepper have always produced very well with no disease. I live in zone 5/6 (western Kentucky).

I have small community garden plots, so I plant them in the same bed, 2 ft. apart. I can do this because I live in a summer-dry climate where the leaves do not get wet. I don't see foliage diseases until after the first big rainstorm in October.

Tomatoes are more susceptible to diseases if they are crowded and if their leaves get wet. If you live in a climate where it rains in the summer, you have to give the plants more space: plant them farther apart, prune some branches so that they are not as crowded and air can move freely through the garden, etc.

My climate is also relatively cool in the summer, so my peppers do better with warmer roots (=planted in containers). Last year I bought a 6-pack of the same pepper. I planted one in a 5-gallon container and the rest at 2 different gardens. The one in the pot was 2-3 times the size of any of the other ones, and it also produced peppers that were 2-3 times larger than any others.

I also plant my tomatoes and peppers near each other. I have a huge garden area, no raised beds, though I love them. I find that my pepper plants do fantastic, if they are allowed to 'touch' when they are full grown. I have no idea why this works, you'd think they'd like space, but they just do better..I do run into the peppers crossing with each other, but the peppers are huge and fully loaded, so, you might try some together and some apart, and see what the diffference is.
Placing marigold plants next to each tomato plant seems to help with the aphid issues, but nothing cures it..if you're very lucky, a bunch of lady bugs will show up and clean them up for you!