Can I add a variable that varies only with time in Least Squares Regression model with a time-fixed effects term? I'm estimating this equation for the trade flows between various countries at time t: ($i$ and $j$ are countries)

The last 3 terms are control variables, one of which is oil prices. Can I add oil prices as a control variable if I already have a time fixed effects term, $ \beta_3 $. Further, can I have country pair fixed effects, $\beta_6$, when I'm also controlling for variables that are fixed for two countries across time (for example, if the two countries were in a colonial relationship in the past, this variable wouldn't change with time and would stay fixed for the two countries for the entire period).
Your model has $\beta_3 * t$, which is a linear time trend, not time dummies. If that's correct, you are controlling for only a linear trend. Because oil prices do not have a perfect linear trend, you can include them.
But I am not sure you really want the linear trend specification instead of time dummies (say, $\beta_{3t}$). For a model with common time effects (year dummies), time series variables are not allowed due to perfect collinearity. Thus, if the time series variables are the key variables, you have a trouble. But if you only want to control for them, you can just omit them because any common time effects will be eliminated by the time dummies.
Country-pair fixed effects are similar. You cannot use time-invariant variables if those fixed effects are included.
One possible solution is to take the correlated random effects approach, but there is the issue of RE $\ne$ FE. You will have to defend your RE model. That's a different issue.