How closely related are modern Italians and ancient Romans?

This question belongs in /r/AskAnthropology where I have a "Population Genetics" flair and could answer your question with flair.

One problem is that the ancient Romans cremated themselves, for the most part. This destroys DNA that could be used for genotyping.

But we can study the distribution of different genetic markers that currently exist in Italy to geographic areas where those markers are present in high concentration.

Here are some peer reviewed literature as evidence

Uniparental Markers of Contemporary Italian Population Reveals Details on Its Pre-Roman Heritage

Y chromosome genetic variation in the Italian peninsula is clinal and supports an admixture model for the Mesolithic–Neolithic encounter This figure from the PLoS paper is a good summary

The left side of the figure shows mitochondrial haplotypes, these are maternal in origin. The right side shows Y chromosome haplotypes, these are paternally inherited.

Firs thing to notice is that it's a distribution of many haplotypes. This suggests many human migration events, which are evident in historical documents and archeology.

A breakdown of the figure as follows:

Mt haplotype H has it's highest percentage in North African populations but is also associated with many European populations suggesting an appearance in the European population around 25,000 years ago.
Mt haplotype U is found throughout Europe, West Asia, North Africa, and India.
Mt J is believed to be originated around West Asia; J is for mitochondrial mother Jasmine.
Y R1b3 is very common in north European populations, but is widespread all over the world. You can see in the figure it's most common in areas where a Celtic/Celto-Germanic identity remains strong. Interesting this is most common in northern Italy.
Y J2 is very common among Mediterranean populations and believed to be originated in the Arabian peninsula. This is commonly referred to the Semitic haplotype and is also found in populations of Ashkenazi Jews.
Y I haplotypes are usually only found in European populations

In short, yes all those populations had a genetic impact on Italy. Due to the lack on genetic material of ancient (pre-Christan) Romans, we may never know for a fact how "closely related" modern Italians are to ancient Romans. Although a paper about the Etruscans and Tuscans suggest that they are more related to each other than other European populations. From this we can extrapolate that the genetic map of Italy has changed in the past and will continue to change, but we can look at current populations and infer human migration events that surprisingly (or not) correlate with historical events, such as the Jewish Diaspora.