Object before the verb 每次地理考试我都不及格
Your question is very interesting.
考試 here is not an object. It belongs to 每次 phrase.
We can simply treat it like this pattern.

每次 + conditional phrase + consequence phrase (usually with 都 or more archaic with 皆 indicating that it is always like that).

This is a common pattern for expressing recurring consequence under certain conditions.
With this pattern, we can break it into two parts.

每次地理考試　我都不及格

More examples

每次大雨　我都繞路 (Heavy rain -&gt; I go for indirect route)
每次地震　我都不在 (Earthquake -&gt; I am not there)
每次閃燈　都是紅色 (Light flashes -&gt; (it is) red in colour)
每次你去澳門　都忘記帶禮物 (You go to Macau -&gt; (you) forget to bring some presents)
每次外婆看醫生　母親都會陪同 (Grandmother visits doctor -&gt; Mother go with her)
每次出戰　皆勝 ((someone) has -&gt; (someone) wins)

Sure you can add 有 in the 每次 phrase.

每次有地理考試　我都不及格

The meaning is the same but there are some subtle grammatical changes.
The interesting thing happens here. Chinese languages do not explicitly show the part of speech like English. Most Chinese speakers cannot tell what the grammatical properties of words is unless it is very simple.  There would be a spectrum of answers because most Chinese grammar books imitates Western grammar books.
Actually there are two layers of grammar of words. First is the natural layer. Some words bear clear property naturally, for example, 踢 (kick) is verb, 紅 (red) is adjective, 葉 (leaf) is noun. But many more words are vague and ambiguous. In order to make sentence comprehensible, second layer let the placement of words in sentence to decide the grammatical properties of in usage. For example, 踢 in 一踢 is a noun because it follows 一  like 一杯, 一場. This overrides its natural property in usage.
每次 is like 每個. There is a noun following 每次. We can find phrase like 每次大雨 and 每次火災. We can conclude that 地理考試 in 每次地理考試 works as noun here. For 每次你去澳門, 你去澳門 is  a phrase functioning as noun as well. So do 每次有地理考試. 有地理考試 functions as noun here.

地理考試 is a noun in 每次地理考試
有地理考試 is a sentence dressing itself as a noun in 每次有地理考試.

Phrase and sentence patterns is very important in Chinese languages. In general, word order dictates the meaning of words and their parts of speech.