If both of these things happen, the price of car tires will go up. Both of the changes that you mention will tend to make the price of car tires go up.
If the price of rubber increases, then (all other things being equal) the price of tires will increase as well. Rubber is, of course, an input for car tires. The price of inputs is a determinant of supply. When the price of inputs increases, supply decreases. When supply decreases, the price of the good or service rises. Therefore, when the price of rubber increases, the price of car tires will increase as well.
When the price of cars goes down, the price of tires will also go up. Cars and tires are what are called complementary goods. That is, the two of them are used together. When the price of one complementary good goes down, demand for the other goes up. If the price of cars goes down, people will buy more cars. If people buy more cars, they will need more tires to go on those cars. This means that the demand for car tires will rise and their price will rise as well.
In these ways, we can see that both of the factors that you mention in this question will tend to push the cost of car tires upward.
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