Actually, without context it is impossible to determine the correct tense in English. Anyway: I'm not going via Tiananmen. I won't be passing through Tiananmen. I presume you are going somewhere, but not via Tiananmen.

1). 首先时态上，你的感觉是正确的。因为那句话只是表明一个事实，不是已经发生的情况。不过我想请教一下Daniel，“我不经过天安门那边”可不可以翻译成：I don't go anywhere close to Tiananmen? Correct me if I'm wrong. 2). “通过” is alternative to “经过” 尤其当后面是跟的抽象名词时。But there's still subtle difference between them. 如果后面是跟的地点，那“通过”强调的是“through”，而”经过“只是”past“。

OK.. In general case, it could be "i won't pass the area around Tiananmen." or " im not gonna go to anywhere close to Tiananmen" . if someone asks you to get her/him something that you can only get in the shops around Tiananmen, but you're not going to get there as usual , you may say" 我(这次)不经过天安门那边。". In past circumstance, the sentence should be "我没经过天安门那边。". if someone has asked your to buy her/him sth., however, you've not gotten because you didn't pass there. you could say "我没经过天安门那边" or "我没去天安门那边".

It simply sounds like somebody is going somewhere, but won't be going via Tiananmen.

there are many translation versions, it depends on how the context is..