In short:
No
Do not trust your conscience.
Regarding this verse above, our hearts and our consciences are the same thing, biblically speaking. They are part of our "soul" (in biblical terms). (See this question for distinction between the soul and the spirit.)
To show that these are the same, I'll go back to the original Greek, so that we can understand the original intention.
Conscience defined
If we look at the Romans 2:14 (from the question) and examine the Greek, we find the word for "conscience" is syneidesis. However, the definition is a bit foggy:
1) the consciousness of anything
2) the soul as distinguishing between what is morally good and bad, prompting to do the former and shun the latter, commending one, condemning the other
a) the conscience
Here, we see that the word can mean either our "consciousness" and our "conscience" (our sense of right and wrong). In context, we can determine that it is the second definition that Paul is intending to use. He's comparing this to a man's heart, so we can safely translate this word as conscience, meaning the second definition.
Heart defined
Romans 2:14 (from the question) uses the word "heart" (as in the laws are written on our hearts). The Greek word for this is kardia
This word clearly references the physical organ. However, it also has a bit of a metaphorical use (which is comparable to its use in English). Ignoring the "organ" definitions, we see:
1) the vigour and sense of physical life
2) the centre and seat of spiritual life
a) the soul or mind, as it is the fountain and seat of the thoughts, passions, desires, appetites, affections, purposes, endeavours
b) of the understanding, the faculty and seat of the intelligence
c) of the will and character
d) of the soul so far as it is affected and stirred in a bad way or good, or of the soul as the seat of the sensibilities, affections, emotions, desires, appetites, passions
When Paul uses the words "heart", he seems to be talking about the second definition: "the center and seat of spiritual life". More specifically, he's referring to the soul as it's affected by morals, emotions, desires, etc.
Conscience = Heart
With these two words defined, we can clearly see that he's using two separate words (one literally and one metaphorically) to illustrate a concept: God's laws are inside of us, written on our hearts, implanted in our conscience. (Note the three "places" in that sentence are really the same concept.)
In the same sense as above...
Do Not Trust Your conscience!
The verse you show in your question is a great example of people who do not have the law. In context of the scripture, this means that they are not Jews. The Law (Old Testament rules and regulations) were not given to the Gentiles, so they do not have any guiding standard to determine right from wrong.
Instead, God has given them their conscience as a guide. He has written the laws on their hearts--implanted them in their consciences. Does this mean that their conscience are infallable? Absolutely not!
While they have the laws written on their hearts, in their humanity, they are fallible (All have sinned, etc.). However, these non-Jews / non-Christians who do not have the law are subject to their consciences instead.
Romans 2:15b (NASB)
...their conscience bearing witness and their thoughts alternately accusing or else defending them
This does not mean that they will be free from sin. However, if they earnestly sought to do what is right in God's eyes, they will be justified even though they did not know what was "right" or "wrong" (per the Law).
Does this mean we can trust our conscience? I would say a huge, incredible No!! The reason is because we have been given explicit guidance on what is right and wrong: specifically, the Bible. The Gentiles in your verse did not know right from wrong, but we do.
That's why we're commanded to test the spirits:
1 John 4:1a (NIV)
Dear friends, do not believe every spirit, but test the spirits to see whether they are from God
Like the Heart/Conscience metaphor above, here Paul tells us to test the "spirits" to determine whether they are from God. We are to determine whether the internal pullings and tugs we hear inside of us (either as self-talk or as emotional affects) are in accordance with God's will. How do we determine God's will? By reading the bible.
If you have a "spirit" telling you to do something contradictory to the Bible, that thought or emotional tug is not from God.
Our hearts are deceitful
One final verse I'll throw in here for all the naysayers:
Jeremiah 17:9 (NIV)
The heart is deceitful above all things
and beyond cure.
Who can understand it?
Do NOT trust your heart.
Summary
1 John 4:1a (NIV)
Dear friends, do not believe every spirit, but test the spirits to see whether they are from God
The idea with this verse is that we should question what goes on inside of us. If we hear something that is telling us that we should go one way or do a given deed, we need to test that guidance to determine whether or not it is from God.