> Question: Do I have to add something to my model to account for the correlation within participant between conditions across trials, and if yes, what?

No, you do not need to. Random intercepts or coefficients by group are equivalent to compound symmetry of the error variance-covariance matrix by group. Therefore, a correlation structure of the residuals is often unnecessary in presence of random effects. Otherwise, the algorithm will be confused about how to split the correlated pattern between random effects and correlated errors, a dilemma analogous to perfect multicollinearity among predictors.

By `(1 + factor1 * factor2 | participant)`, you will have 4 + 3 + 2 + 1 = 10 extra parameters to estimate for the variance-covariance (variance component) matrix of random effects. Because all random effects are grouped by the participant, the error term (which includes random intercepts and random coefficients) among 24 measurements within each participants are already correlated, which results in an intra-class correlation (ICC) at $\sigma^2_\text{random} / (\sigma^2_\text{random} + \sigma^2_\text{residual})$. Nevertheless, this 10-element matrix is unrestricted (a general positive-definite symmetric matrix) might be too complex. If you see any component to be close to zero, you can restrict this variance component matrix to follow a simpler pattern (e.g., diagonal for only 4 parameters). 

In your study, each participant experienced four conditions out of two two-level factors, levels A and B in `factor1` and levels 1 and 2 in `factor2`. Under each condition, a participant was measured six times. Therefore, each participants have 24 measurements. Just like if someone measures my height six times, these measurements are independent although they are all of the same person. 

Theoretically, the trials will be independent if their error term (the difference between predicted and observed `response`) is not related to each other. There is no way to prove it because the error term cannot be directly observed, but this error independence needs to be justified logically. Does one overestimation in one trial leads on average an overestimation or underestimation in the next trial? 

Practically, modeling correlation across trials within the same participant requires prespecified patterns of the errors. Are the trials done by different surveyors, scales, or equipment, so they can be grouped by `Surveyor 1` ... `Surveyor n`? Are the trials done sequentially on `Day 1`, `Day 2`, ..., `Day 6`? Did the trials take place at specific locations that can be geographically grouped into `Location A`, `Location b`, ...? If the study did not record any information regarding how different trials might be associated with each other, a researcher cannot retrieve the error-correlation pattern by trial even if one suspects any. 

The package {nlme} allows modeling random intercepts and coefficients along with AR process of the error term, but it may not be necessary or significant for certain data. In the `sleepstudy` data for example, if the variation by participant is controlled for, the residuals that represent the difference between predicted and observed reaction time may not present enough temporal correlation to necessitate error-correlation consideration. You can also use the package `glmmTMB` for some special error-correlation patterns.